THE CONVEX REAL PROJECTIVE MANIFOLDS AND
ORBIFOLDS WITH RADIAL ENDS I: THE OPENNESS OF
DEFORMATIONS
SUHYOUNG CHOI
Abstract. A real projective orbifold is an n-dimensional orbifold modeled on RP n
with groups PGL(n + 1, R). We concentrate on orbifolds with a compact codimension
0 submanifold whose complement is a union of neighborhoods of ends, diffeomorphic
to (n − 1)-dimensional orbifolds times intervals. A real projective orbifold has radial
end if each of its end is foliated by projective geodesics concurrent to each other.
It is said to be convex if any path can be homotoped to a projective geodesic with
endpoints fixed. A projective structure sometimes admit deformation to inequivalent
parameters of real projective structures. We will prove that local homemorphism
between the deformation space of projective structures on such an orbifold with radial
ends with various conditions with the representation space of the fundamental group
with corresponding conditions. We will use a Hessian argument to show that a small
deformation of a real projective orbifold with ends and without essential annuli will
remain properly and strictly convex in a generalized sense if so is the beginning real
projective orbifold provided that the ends behave in a convex manner. Here, we have
to restrict each end to have a fundamental group isomorphic to a finite extension of
a product of hyperbolic groups and abelian groups. The understanding of the ends
is not accomplished in this paper as this forms an another subject. We will prove
the closedness of the convex real projective structures on orbifolds with irreducibilty
condition. One theorem of note is that a convex irreducible real projective orbifold
with radial ends with some condition is relatively hyperbolic if and only if it is strictly
convex with respect to ends, generalizing the result of Benoist for closed orbifolds.
Contents
List of Figures
1. Introduction
2. Preliminary
2.1. Orbifolds with ends
2.2. Geometric structures on orbifolds with ends
2.2.1. Affine and real projective manifolds
2.2.2. Affine suspension constructions
3. The local homeomorphism theorems
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12
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Date: January 14, 2009.
1991 Mathematics Subject Classification. Primary 57M50; Secondary 53A20, 53C15.
Key words and phrases. geometric structures, real projective structures, SL(4, R), representation
of groups.
This work was supported by the National Research Foundation of Korea (NRF) grant funded by
the Korea government (MEST) (No. 2010-0000139).
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3.1. The end fundamental group condition
3.2. Local homeomorphism theorems
3.3. The proof of the local homeomorphism theorems
3.3.1. Affine structures
3.3.2. Real projective structures
4. Convexity
4.1. Convexity and convex domains
4.2. Definitions associated with ends
4.3. Properties of ends
4.3.1. Properties of horospherical ends
4.3.2. The properties of lens-shaped ends
4.3.3. Examples
4.4. The Hilbert metric on O.
4.5. Strict IPC-structures and the relative hyperbolicity of the metric spaces
4.5.1. The convex hulls of ends
4.5.2. Bowditch’s method
4.5.3. Converse
4.5.4. The relative hyperbolicity of the Hilbert metric of IPC-structures
5. Openness of the convex structures
5.1. Proof of the convexity theorem
6. The closedness of convex real projective structures
7. Examples
7.1. Main examples
7.2. Nonexistence of reducible holonomy
References
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List of Figures
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The universal covers of horospherical and lens shaped ends. The radial lines
form cone-structures.
The figure for Lemma 4.14.
The structure of a lens-shaped end.
The shortest geodesic m to a geodesic l.
The diagram for Theorem 4.27.
The diagram for Lemma 5.6.
A convex developing image example of a tetrahedral orbifold of orders
3, 3, 3, 3, 3, 3
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1. Introduction
Recently, there were many research papers on convex real projective structures on
manifolds. (See the work of Goldman [29], Choi [16], [17], Benoist [5], Inkang Kim [45],
Cooper, Long, Thistlethwait [24], [25] and so on.) One can see them as projectively
2
flat torsion-free connections on manifolds. Topologists will view them as an n-manifold
with maximal atlas of charts to an n-dimensional real projective space with transition
maps projective transformations. Hyperbolic and many other geometric structures will
induce canonical real projective structures. Sometimes, these can be deformed to real
projective structures not arising from such constructions. We think that these results
are useful in studying linear representations of discrete groups and so on as the discrete
group representation and deformations form very much mysterious subjects still.
The deforming a real projective structure to an unbounded situation results in the
action of the fundamental groups on buildings which hopefully will lead us to some
understanding of manifolds in particular in dimension three as indicated by many
people including Cooper, Long, and Thistlethwait. Also, understanding when such
deformations exist seems to be an interesting question from the representation theory
point of view.
However, the manifolds studied are closed ones so far. The work easily generalizes
to closed orbifolds. We hope to generalize this to noncompact ones with conditions
on ends. These are n-orbifolds with compact suborbifolds whose complements are
diffeomorphic to intervals times (n − 1)-dimensional orbifolds. Such orbifolds are said
to be topologically tame orbifolds. Because of this, we can associate an (n − 1)-orbifold
at each end and we define an end fundamental group as the fundamental group of the
orbifold. The orbifold is said to be the end orbifold. We also put the condition on end
neighborhoods being foliated by radial lines. Of course, this is not the only natural
conditions. We plan to explore the other conditions in some other occasions.
Note that we did studied such orbifolds in [19] where the orbifolds were of Coxeter
type with ends. These have convex fundamental polytopes and are easier to understand.
Thus, this paper generalize the results there.
D. Cooper and S. Tillman studied a complete hyperbolic 3-orbifold obtained from
gluing a complete hyperbolic tetrahedron. The deformation exists and can be solved.
However, the convexity of the result was the main question that arose. We will try
to answer this in this paper. They are also studying the same subject as us and have
claimed some very interesting analogous results but we have no idea of the precise
nature of their work as it is not available presently.
In general, the theory of real projective structures on manifolds with ends are not
studied very well. We should try to obtain more results here and find what are the
appropriate conditions. This seems to be also related to how to make sense of the
topological structures of ends in many other geometric structures such as symmetric
spaces and so on for which there are many theories currently.
Let O be a noncompact topologically tame n-orbifold. The orbifold boundary is assumed empty. A real projective orbifold is an orbifold with a geometric structure modelled on (RP n , PGL(n + 1, R)). A real projective orbifold also has notion of projective
geodesic as given by local charts and has a universal cover Õ with a deck transformation group π1 (O) acting on it. The underlying space of O is homeomorphic to the
quotient Õ/π1 (O). A real projective structure on O gives us a so-called development
pair (dev, h) where dev : Õ → RP n is an immersion, called the developing map, and
h : π1 (O) → PGL(n + 1, R) is a homomorphism, called a holonomy homomorphism,
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satisfying dev ◦ γ = h(γ) ◦ dev for γ ∈ π1 (O). The pair (dev, h) is determined only
up to the action g(dev, h(·)) = (g ◦ dev, gh(·)g −1 ) and any chart in the atlas extends
to a developing map. (We can make the boundary be nonempty however with more
stronger conditions. Unfortunately, these are not in the researched topics. We will
work on this later.)
A real projective orbifold has radial end if each end has end neighborhood foliated
by properly imbedded projective geodesics that extend to concurrent geodesics for each
chart and nearby leaf-geodesics develop to open geodesics in RP n ending the common
point of concurrency. Two such radial foliations are compatible if they agree outside
any compact subset of the orbifold. A radial foliation marking is the compatibility
class of a radial folations. A real projective orbifold with radial end marks is such a
topologically tame orbifold with end neighborhood radial foliation marking. In this
case, each end has a neighborhood diffeomorphic to a closed orbifold times an interval.
This orbifold is independent of the choice of such a neighborhood and it said to be
the end orbifold associated with the end. Its fundamental group is isomorphic to the
fundamental group of an end neighborhood, and the end orbifold can be imbedded
transversal to the radial foliation. We also call the component of the inverse image of
the end neighborhood in Õ an end neighborhood. The radial foliation has a transversal
real projective structure and hence the end orbifold has a unique induced real projective structure of one dimension lower as the concurrent lines to a point form RP n−1
and the real projective transformation fixing a point correspond to a real projective
transformation of RP n−1 . The end orbifold is convex if O is convex. If the end orbifold
is properly convex, then we say that the end is an properly convex end.
We will assume that our orbifolds are usually topologically tame and with radial
ends with markings.
We define Def(O) as the deformation space of real projective structures on O with
end marks; more precisely, this is the set of real projective structures µ on O so
that two structures µ1 and µ2 are equivalent if there is an isotopy i on O so that
i∗ (µ2 ) = µ1 where i∗ (µ2 ) is the induced structure from µ2 by i and i preserves the
radial end foliation markings. We define Def E (O) to be the subspace of Def(O) with
radial ends. Def RP n ,E,ce (O) is the subspace of real projective structures so that induced
real projective structures with admissible ends.
The representation space rep(π1 (O), PGL(n + 1, R)) is the quotient space of the
homomorphism space Hom(π1 (O), PGL(n + 1, R))/PGL(n + 1, R) where PGL(n + 1, R)
acts by conjugation h(·) 7→ gh(·)g −1 . We define repE (π1 (O), PGL(n + 1, R)) to be the
subspace of representations where the restricted representation to each end fundamental
group has nonzero common eigenvectors.
An end fundamental group condition is defined in Section 3.1. Loosely speaking, this
condition is one where if a representation of an end fixes a point of RP n , then it fixes
a unique one.
The following map is induced by sending (dev, h) to the conjugacy class of h:
Theorem A . Let O be a noncompact topologically tame n-orbifold. Suppose that O
has the end fundamental group conditions. Then the map hol : Def E (O) → repE (π1 (O), PGL(n+
1, R)) is a local homeomorphism.
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The complement of a codimension-one subspace of RP n can be identified with an
affine space Rn where the geodesics are preserved and the group of affine transformations of Rn is obtained by restricting the group of projective transformations of
RP n fixing the subspace. We call the complement an affine patch. It has a geodesic
structure of a standard affine space. A convex domain in RP n is a convex subset of
an affine patch. A properly convex domain in RP n is a convex domain contained in a
precompact subset of an affine patch.
An important class of real projective structures are so-called convex ones: i.e., where
any arc in O can be homotoped with endpoints fixed to a straight geodesic. If the orbifold has a convex structure, it is covered by a convex domain Ω in RP n . Equivalently,
this means that the image of the developing map dev(Õ) is a convex domain. Here we
may assume dev(Õ) = Ω, and O is projectively diffeomorphic to Ω/h(π1 (O)). In our
discussion, since dev is an imbedding, Õ will be regarded as an open domain in RP n
and π1 (O) a subgroup of PGL(n + 1, R), which simplify our discussions.
An IPC-structure or an irreducible properly-convex real projective structure on an norbifold is a real projective structure so that the orbifold with radial end is projectively
diffeomorphic to a quotient orbifold of a properly convex domain in RP n by a discrete
group of projective automorphisms fixing no proper subspace of RP n up to finite index.
Define bdA for a subset A of RP n to be the topological boundary in RP n and define
∂A for a manifold or orbifold A to be the manifold or orbifold boundary. The closure
Cl(A) of a subset A of RP n is the topological closure in RP n . A subdomain K of RP n
is said to be horospherical if it is strictly convex and the boundary ∂K is diffeomorphic
to Rn−1 and bdK − ∂K is a single point. K is lens-shaped if it is a convex domain
and ∂K is a disjoint union of two smoothly embedded (n − 1)-cells not containing any
straight segment in them. A cone is a domain in RP n whose closure in RP n has a point
in the boundary, called a cone-point, so that every other point has a segment contained
in the domain with endpoint the cone point and itself. A cone-over a lens-shaped
domain A is a convex submanifold that contains a lens-shaped domain A of the same
dimension and is a union of segments from a cone-point 6∈ A to points of A and the
manifold boundary is one of the two boundary components of A. Alternatively, we can
take the union of segments from the cone-point and the points of the component of ∂K
further away from the cone-point so that the boundary is the component. It is radial
set. A lens-cone is the union of the segments over a lens-shaped domain. A lens is
the lens-shaped domain A, not determined uniquely by the lens-cone itself. A totallygeodesic subdomain is a convex domain in a hyperspace. A cone-over a totally-geodesic
domain A is a cone over a point x not in the hyperspace.
In general, a join of two convex sets C1 and C2 is the union of segments with end
points in C1 and C2 respectively and is denoted by C1 + C2 in this paper. We can
generalize to the sum of n sets C1 , . . . , Cn . A cone-over a joined domain is a one
containing a joined domain A and is a union of segments from a cone-point 6∈ A to
points of A where the cone point is given by V 0 ∩ V 00 .
Let E be an (n−1)-dimensional end orbifold, and let µ be a holonomy representation
π1 (E) → PGL(n + 1, R) fixing a point x. Suppose that µ(π1 (E)) acts on a lens-shaped
domain K in RP n not containing x with boundary a union of two open (n − 1)-cells
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A and B and π1 (E) acts properly on A and B with compact Hausdorff quotients
and the cone of K over x exists. Then µ is said to be a lens-shaped representation
for E with respect to x. µ is a totally-geodesic representation if µ(π1 (E) acts on a
cone-over a totally-geodesic subdomain with a cone-point x. If µ(π1 (E)) acts on a
horospherical domain K, then µ is said to be a horospherical representation. In this
case, bdK − ∂K = {x}. If µ(π1 (E)) acts on a joined domain, its cone-point and
associated subspaces Vi0 s and Vj00 s, then µ is said to be a joined representation.
We will only study ends that are horospherical or lens-shaped for simplicity. Consider an properly convex end of an orbifold with radially foliated end neighborhood.
An end is lens-shaped, lens-shaped, or totally geodesic according to whether it has a
neighborhood isomorphic to the cone over such domains with vertex the end vertex
with radial foliation going to lines from the end vertex. The end vertex of an end has
vertex eigenvalue 1 if the holonomy of each element of the end vertex group act with 1
as the eigenvalue of the end vertex.
We will see that nontrivially joined lens-shaped ends implies that the end is totally
geodesic by Theorem 4.16 when our ambient orbifold is not virtually reducible. Also,
totally geodesic end with vertex eigenvalue 1 conditions is a lens-shaped ends. (See
Section 4.3.3 and Theorem 7.1.) Also, in some cases with conditions on the eigenvalues
of the end fundamental group holonomy, we obtain lens-shape ends for properly convex
ends. However, we need some restrictions on the ambient orbifold group However,
there are examples of nonjoined lens-shaped domain that is not totally geodesic in the
irreducible case using “bending”. We will discuss these topics in an another paper [22].
In fact, given relative hyperbolicity of the fundamental group with respect to the end
fundamental groups, we will attempt to show that we always have lens-shaped ends or
modify the ends to be lens-shaped for properly convex ends. In fact, we plan to study
much of the issues on ends there.
In this paper, lens-shapedness play essential role in many parts of the proofs so that
we believe this must be an essential necessary feature of any theory to be developed.
However, without hyperbolicity, we believe that the restrictions are violated easily
as can be seen by the fact that 2-dimensional real projective annulus with principal
boundary can be deformed to ones that are not. We do believe these warrant future
studies and is not easily doable in a near future by us.
We study lens-shaped ends because they should be generic and they are analogous to
quasi-Fuchsian manifolds in hyperbolic manifold theory and other type of ends are not
very understandable for us yet. Since the theory of quasi-Fuchsian groups are large,
we question that the theory for convex radial ends could be parallel and extensive.
There are examples of properly convex ends that are not lens-shaped or horospherical
in any dimension using bending as above. Non-properly convex ends that are not
horospherical is not classified yet and it is a future topic.
Note that lens-shaped ends are stable under the deformations of the fundamental
groups; that is, a small deformation of the representation preserves this property. However, this is not a closed condition just as in the two-dimensional cases. (See [16] and
[17]). Horospherical ones may not be stable. See Theorem 4.17. (Note that these
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representations are not in hyperconvex components of Labourie [42] and [43]. because
of the common fixed points.)
If every subgroup of finite index of a group Γ has a finite center, Γ is said to be
virtual center-free group or a vcf-group. An admissible group is a finite extension of a
finite product of Zl × Γ1 × · · · × Γk for infinite hyperbolic groups Γi where l ≥ k or
k = 1 holds and l + k ≤ n holds. (See Section 4.3 for details. l ≥ k follows from
the result of Benoist discussed their. We have k = 1 and l = 0 if and only if the
end fundamental group is hyperbolic. ) (For example, if our orbifold has a complete
hyperbolic structure, then end fundamental groups are virtually free abelian.) If we
understand the ergodicity properties of vcf-groups, we would be able to expand the
class of groups of Γi . However, we are not able to understand the properties yet.
We assume that the fundamental group of each end orbifold is an admissible product
group. These are the main possibility in an inductive sense we can consider when we
are studying the orbifolds with end real projective orbifold structures that deform to
ones of the properly convex (n−1)-dimensional real projective orbifolds by Proposition
4.4 (see Benoist [6]). (Unfortunately, we are unable to study other groups. For this, we
need to classify convex but not properly-convex real projective n-manifolds as occurring
in [14]. Other possibilities to restrict the groups to be nilpotent but the same difficulty
is there.)
We say that the radial end is admissible if either it has a neighborhood whose universal cover is a horospherical domain or is a cone over a lens-shaped domain for the
corresponding representation of π1 (E) whose quotient is diffeomorphic to E × [0, 1] and
form the neighborhood of E for a corresponding end orbifold E. We require that the
cone-point has to correspond to the end of the radial lines for the given radial end.
More precisely, this means that in the developing image of the universal cover of the
open neighborhood of the radial end the radial lines are contained in the segments
from the cone-point. We do not require the closures of the developing images are also
strictly convex.
We will also say that an admissible end is hyperbolic if the end fundamental group
is hyperbolic and is Benoist if k = l ≥ 1. Benoist or hyperbolic ends are said to be
permanantly properly convex. (See Section 4.3.2.)
We cannot classify all possible the radial ends of real projective manifolds yet. This
is an unresolved subject. But mostly, we believe that some forms of joins of lensshaped and horospherical ends, i.e., joined ends, are essentially everything that we
could understand. We think that there are more general types of ends but if we
restrict the end groups to be nilpotent, then there might be a good chance to classify
them as nilmanifolds are classified by Benoist [11] as a future project. Also, some of the
solutions of the Auslander conjecture is necessary here, and these topics are of different
nature from the geometric arguments here. We feel that important geometric issues
are mostly done in this paper and the end theory only present algebraic difficulties.
We will classify lens-shaped ends in Subsection 4.3.2 and show that they are strict
lense-shaped, i.e., the complement of the boundary is nowhere dense. The lens-shaped
ends are introduced as naturally from two-dimensional analogies. We believe that they
are essential in studying the radial ends and without this condition, we might loose too
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Figure 1. The universal covers of horospherical and lens shaped ends.
The radial lines form cone-structures.
much control. The condition generalizes that of Goldman [29] for convex real projective
surfaces with totally geodesic boundary. Without this condition, we can study these
surfaces as well [16] and [17]; however, there are some degeneracies.
Let O be a tame n-orbifold where end fundamental groups are admissible. We define
repE,ce (π1 (O), PGL(n + 1, R)) to be the subspace of
repE (π1 (O), PGL(n + 1, R))
where each end is realized as admissible end of some real projective orbifold mapping
into O as an end. (The realization is essentially unique by Remark 4.18.) We define repiE,ce (π1 (O), PGL(n + 1, R)) to be the subspace of irreducible representations in
repE,ce (π1 (O), PGL(n + 1, R)).
The orbifold O has no essential homotopy annulus if no element of the fundamental
group one end of O is homotopic to another element of the fundamental group of an
end of O and the homotopy annulus cannot be pushed into one end. (This is from
3-manifold topology).
Suppose that O has an IPC-structure. If every straight arc in the boundary of the
domain Õ is contained in the closure of a component of a chosen equivarient set of end
neighborhoods in Õ, then O is said to be strictly convex with respect to the collection
of the ends. And O is also said to have a strict IPC-structure with respect to the
collection of the ends.
A convex end fundamental condition is defined in Section 3.1. Loosely speaking, this
condition is one where if a representation of an end of convex orbifolds with radial
admissible ends fixes a point of RP n , then it fixes a unique one. This is more general
than a mere end fundamental condition.
We define Def iE,ce (O) to be the deformation space of real projective structures with
admissible ends and irreducible holonomy and define CDef E,ce (O) to be the deformation
space of IPC-structures with admissible ends.
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Theorem B . Let O be a noncompact topologically tame n-orbifold with admissible
ends. Suppose that O satisfies the convex end fundamental group conditions. Then
• In Def iE,ce (O), the subspace CDef E (O) of IPC-structures is open.
• Suppose further that π1 (O) contains no notrivial nilpotent normal subgroup.
The deformation space CDef E,ce (O) of IPC-structures on O maps homeomorphic to a component of repiE,ce (π1 (O), PGL(n + 1, R)).
Theorem 5.1 and Theorem 6.1 proves this and following theorems.
Theorem C . Let O be a strict IPC noncompact topologically tame n-dimensional
orbifold with admissible ends and convex end fundamental group condition. Suppose
also that O has no essential homotopy annulus or torus. Then
• π1 (O) is relatively hyperbolic with respect to its end fundamental group.
• In Def iE,ce (O), the subspace SDef iE (O) of strict IPC-structures with respect to
all of the ends is open.
• Then the deformation space SDef E,ce (O) of strict IPC-structures on O with
respect to all of the ends maps homeomorphic to a component of
repiE,ce (π1 (O), PGL(n + 1, R)).
We also showed that an IPC-orbifold O with admissible end is strictly IPC iff π1 (O)
is relatively hyperbolic with respect to its end fundamental groups. (See Theorems
4.26 and 4.27.)
We remark that there should be another approaches to these theorems using the
work of J. Lee [44]. However, such a theory should have different assumptions on the
fundamental domains and the groups and so on.
In Section 2, we give elementary definitions of geometric structures, real projective
and affine structures, parallel and radial ends, and so on. We also study an affine
suspension, a method to obtain an affine structure from a real projective structure.
In Section 3, we prove the local homeomorphism theorem, i.e., Theorem A; that
the maps denoted by hol send the deformation spaces to the representation spaces in
locally homeomorphic fashion. In our paper, we need to consider radial end structures
and corresponding holonomy conditions. There are end fundamental group conditions,
which arise naturally. We will prove the theorem for the affine structure and change it
to be applicable to real projective structures. The methods are similar to what is in [18].
Here, we need to have continuous section of eigenvectors in the end holonomy group.
Finally, we transfer the theorem to the real projective cases using affine suspensions.
In Section 4, we discuss convexity and define irreducible convex real projective structures on orbifolds. We discuss the decomposition results of Benoist on reducible closed
real projective (n − 1)-orbifolds. We discuss the ends of orbifolds. We discuss horospherical ends and lens-shaped ends and their properties and various facts concerning
their existence, stability, and examples and so on.
We define irreducible properly convex real projective structures or IPC-structures
on orbifolds. This is a convex projective structure whose developing image is properly
convex and the holonomy group do not act on a proper subspace. We define strict IPCstructures also. If O has no essential homotopy annulus, then we show that the convex
hull of end neighborhoods are convex and cocompact. We show that an IPC-orbifold
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has relatively hyperbolic fundamental group with respect to its end fundamental group
if and only if the IPC-orbifold is strictly IPC. (Theorems 4.26 and 4.27.)
In Section 5, we prove that if ends of an orbifold are admissible, then the deformation of (resp. strict) IPC-structures will remain (resp. strict) IPC-structures under
irreducibility conditions, i.e., Theorem B. The proof is divided into two: First, we
show that there is a Hessian metric and under small perturbations of the real projective structures, we can still find a nearby Hessian metric. Basically, we find that
the Koszul-Vinberg functions of the affine suspensions change by very small amounts.
Second, the Hessian metric and the boundary orbifold convexity assumption imply
convexity.
In Section 6, we show that the deformation space of (resp. strict) IPC-structures
maps homeomorphic to an open and closed subset of repiE,ce (π1 (O), PGL(n + 1, R)) the
subspace of irreducible representations.
We conjecture that we can remove the irreducibilty condition in the representation
space in most cases. That is, we can assume that all holonomy representations are
irreducible for strict IPC-structures on sufficiently complicated orbifolds.
In Section 7, we discribe some examples where the theory is applicable. This include
the examples of S. Tillman and a double of a tetrahedron reflection group of all order 3.
Furthermore, for these examples, we can remove the irreducibilty condition of Theorem
C since the tetrahedra in this examples can be controlled.
We thank Yves Benoist, Yves Carrière, Daryle Cooper, Kelly Delp, Craig Hodgson,
Misha Kapovich, William Goldman, Stephan Tillman, Hyam Rubinstein for many
discussions. This paper began from a discussion the author had with Stephan Tillman
on his construction of a parameter of a real projective structures on small complete
hyperbolic 3-orbifold. Unfortunately, we could not continue the collaboration due to
difference in approaches. Many helpful discussions were discussed with Craig Hodgson
and we hope to publish the resulting examples in another paper. We thank Yves
Benoist for many of his results without which we could not have written this paper. I
also thank Misha Kapovich for many technical discussions and help.
We also think that various other forms of the deformation theory of radial ends could
exist. We hope to participate and work with anyone interested in this topic. Somehow,
this theory should be related to other type of end theory occuring in other types of
geometric structures.
2. Preliminary
2.1. Orbifolds with ends. An n-dimensional orbifold is a Hausdorff space with orbifold structures. An orbifold structure is given by a fine covering by open sets with
charts that are quotient maps of open subsets of Rn by finite group of transformations
so that an inclusion map of two open sets induces an inclusion map of the open sets
in Rn with an injection of the groups determined up to conjugations. The orbifolds
enjoy much of the properties of manifolds. As an example, a manifold is an orbifold
in the sense that the manifold is covered by open sets with trivial groups. An orbifold
O often has a simply-connected manifold as a covering space. In this case the orbifold
is said to be good. We will assume this always for our orbifolds. Such a covering Õ
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is unique up to covering equivalences and is said to be the universal cover. There is a
discrete group π1 (O) acting on the universal cover so that we recover O as a quotient
orbifold Õ/π1 (O), where π1 (O) is said to be the (orbifold) fundamental group of O.
The local group of a point of Õ is the direct limit of the group acting on the model
neighborhoods ordered by the inclusion maps. It is well-defined up to local orbifold
isotopy conjugations.
An end neighborhood of an orbifold is a component of the complement of a compact
subset of an orbifold. The collection of the end neighborhood is partially ordered by
inclusion maps. An end is an T
equivalence classes of sequences of end neighborhoods
Ui , i = 1, 2, ..., Ui ⊃ Ui+1 and i=1,2,.. Cl(Ui ) = ∅. Two such sequences Ui and Vj are
equivalent if for each i, there exists j, j 0 such that Ui ⊃ Vj and Vi ⊃ Uj0 .
We study orbifolds with properly foliated ends, which means that each end of the
orbifold has an end neighborhood foliated by properly imbedded real lines and local
group associated with each point of a leaf is locally conjugate by some leaf preserving
flows. Here we require that the leaves develop to concurrent geodesics and nearby leafgeodesics develop to open geodesics in RP n ending at the common point of concurrency.
Two foliations of end neighborhoods are compatible if they agree on a smaller end
neighborhoods. The compatibility class of ends are said to be foliation markings. We
also restrict to a topologically tame orbifold with finitely many ends. This means that
the orbifold is a union of a compact orbifold with finitely many boundary orbifolds and
boundary orbifolds times half open real lines. Also, our orbifold can be compactified
by adding finitely many points corresponding to the ends. We will also assume that
our orbifolds always have foliation markings.
2.2. Geometric structures on orbifolds with ends. Let G be a Lie group acting
on a n-dimensional manifold X. For examples, we can let X = Rn and G = Aff(Rn )
for the affine group Aff(Rn ) = GL(n, R) · Rn , i.e., the group of transformations of
form v 7→ Av + b for A ∈ GL(n, R) and b ∈ Rn . Or we can let X = RP n and
G = PGL(n + 1, R), the group of projective transformations of RP n acting up to
scalar.
Having an (X, G)-structure on an orbifold O means that there are charts from open
subsets of X with finite subgroups of G acting on them and the inclusions always
induces restrictions of elements of G in open subsets of X. This is equivalent to saying
that the orbifold O has some simply connected manifold cover Õ with an immersion
D : Õ → X and the fundamental group π1 (O) acting on Õ properly discontinuously so
that h : Γ → G is a homomorphism satisfying D ◦ γ = h(γ) ◦ D for each γ ∈ Γ. Here,
Γ is allowed to have fixed points. (We shall use this second definition here.) (D, h(·))
is called a development pair and for a given (X, G)-structure, it is determined only up
to an action (D, h(·)) 7→ (k ◦ D, kh(·)k −1 ) for k ∈ G. Conversely, a development pair
completely determines the (X, G)-structure.
An isotopy of an orbifold O is a map f : O → O with a map F : O × I → O so
that Ft : O → O for Ft (x) := F (x, t) every fixed t is an orbifold diffeomorphism and
F0 is the identity and f = F1 . Given an (X, G)-structure on another orbifold O0 , any
orbifold diffeomorphism f : O → O0 induces a (X, G)-structure pulled back from O0
which is given by using the local models of O0 for preimages in O. The deformation
11
space Def X,G (O) of the (X, G)-structures is the space of all (X, G)-structures on O
quotient by the isotopy pullback actions.
This space can be thought of as the space of pairs (D, h) with C r -topology for r ≥ 1
and equivalence relation generated by the isotopy relation (D, h) ∼ (D0 , h0 ) if D0 = D◦ι
and h0 = h for a lift ι of an isotopy and the second conjugacy one (D, h) ∼ (D0 , h0 ) if
D0 = k ◦ D and h(·) = kh(·)k −1 for k ∈ G.
2.2.1. Affine and real projective manifolds. An affine orbifold is an orbifold with a
geometric structure modelled on (Rn , Aff(Rn )). An affine orbifold has notion of affine
geodesics as given by local charts. Recall that a geodesic is complete in a direction if
the geodesic parameter is infinite in the direction. An affine orbifold has parallel end if
each end has end neighborhood foliated by properly imbedded affine geodesics parallel
to each other in charts and each leaf is complete in the end direction. We obtain
a smooth complete vector field XE in a neighborhood of E for each end following
the affine geodesics, which may not be locally parallel. We denote by XO the vector
field partially defined on O by taking union of vector fields defined on some mutually
disjoint neighborhoods of the ends. Note that the oriented direction of the parallel end
is uniquely determined in the developing image of a component of the inverse image of
an end neighborhood in the universal cover of O. Finally, we put a Riemannian metric
on O so that for each end there is an open neighborhood where the metric is invariant
under the flow generated by XO . Note that such a Riemannian metric always exists.
We have SL± , (n + 1, R) = GL(n + 1, R)/R+ , i.e, two linear maps are equivalent if
and only if they differ by a positive scalar. Then this group acts on Sn to be seen as a
quotient space of Rn+1 − {O} by the equivalence relation v ∼ w if v = sw for a positive
scalar s ∈ R+ . Note that the notion of geodesics are defined as in the projective
geometry: they correspond to arcs in great circles in Sn . (Sn , SL± (n + 1, R))-structures
on O is said to be an oriented real projective structure on O. We define Def Sn (O) as
the deformation space of (Sn , SL± (n + 1, R))-structures on O.
An affine or projective orbifold is triangulated if there is a smooth imbedded cycle
consisting of geodesic simplicies on the compactified orbifold relative to endpoints. It is
properly triangulated if each of the interiors of simplicies in the cycle imbedded to one
of same dimension and the local group at each point of the image is locally conjugate.
2.2.2. Affine suspension constructions. Recall that Sn covers RP n by a double covering map q and linear transformations SL± (n + 1, R) of determinant ±1 maps to the
projective group PGL(n + 1, R) by a double covering map q̂. Note that SL± (n + 1, R)
acts on Sn lifting the projective transformations. Given (D, h) of a real projective
orbifold O, we can find a lift D0 : Õ → Sn since Õ is a simply connected manifold and
h lifts to h0 : π1 (O) → GL(n + 1, R).
An affine orbifold is radiant if h(Γ) fixes a point in Rn . A real projective orbifold
O can be made into a radiant affine n + 1-orbifold by taking Õ and D0 and h0 : Define
D00 : Õ × R+ → Rn+1 by sending (x, t) to tD0 (x). For each element of γ ∈ π1 (O), we
define the transformation γ 0 on Õ × R+ by γ 0 (x, t) = (γ(x), ||h0 (γ)(tD0 (x))||). Also,
there is a transformation Ss : Õ × R+ → Õ × R+ sending (x, t) to (x, st) for s ∈ R+ .
Thus, Õ × R+ /hS2 , π1 (O)i is an affine orbifold with the fundamental group isomorphic
12
to π1 (O)×Z where the developing map D is given by D00 the holonomy homomorphism
is given by h0 and sending the generator of Z to S2 . We call the result the affine
suspension of O, which of course is radiant. (See Barbot [1] and Choi [20] also.)
A special affine suspension is an affine suspension using the lift h0 : π1 (O) → SL± (n+
1, R), which always exists.
Note that the affine space Rn is a dense open subset of RP n which is a complement
of (n − 1)-dimensional projective space RP n−1 .
Under cone-construction, a projective (n − 1)-orbifold has radial end if and only if
the affine n-orbifold radially suspended from it has a parallel end. This can be seen
by taking lines in the parallel class of the concurrence point at infinity RP n−1 of Rn
in RP n . Also, if a projective (n − 1)-orbifold is (resp. properly) triangulated, then the
radially suspended affine n-orbifold is also (resp. properly) triangulated.
3. The local homeomorphism theorems
3.1. The end fundamental group condition. Given the an affine orbifold O satisfying our end conditions, we define a subspace HomE (π1 (O), Aff(Rn )) of
Hom(π1 (O), Aff(Rn ))
to be the subspace where each end has neighborhood U and the representation h
restricts to π1 (U ) sends elements to affine transformations with linear parts with at
least one eigenvector. This involves a choice of π1 (U ) in π1 (O) as one can compose
with a path in O. Given an end neighborhoods U , the collection of π1 (U ) is also form
a directed set so that if V ⊂ U , then π1 (V ) → π1 (U ). Let us take the inverse limit of
π1 (U ) and call the result π1 (E). Actually, π1 (E) will equal π1 (S) for a suborbifold S
imbedded in U . The restriction h|π1 (E) has an eigenvector.
Let e be the number of ends of O. Let U be an open subspace of
HomE (π1 (O), Aff(Rn ))
invariant under the conjugation action so that one can choose a continuous section
sU : U → Rne sending a holonomy to a common nonzero eigenvector of π1 (E) for the
choice of the fundamental group π1 (E) for every end E as a subgroup of π1 (O) and
sU satisfies sU (gh(·)g −1 ) = gsU (h(·)) for g ∈ Aff(Rn ). There might be more than one
choice of section in certain cases. sU is said to be the eigenvector-section of U.
There is also an important end fundamental group condition: We say that U and
π1 (O) has the unique fixed direction property for holonomy homomorphisms π1 (E) →
Aff(Rn ) arising from U if the linear parts of each holonomy element of π1 (E) has a
nonzero eigenvector, then it is the nonzero eigenvector unique up to scalar multiplications by some unspecified conditions fixed for U by some unspecified conditions.
Actually, it is not a purely group condition but a geometric condition. In fact, it
might be possible that such a condition holds for a component of representation space
but not for some other subsets of HomE (π1 (O), Aff(Rn )). In such cases, our results are
valid for the components where the conditions hold.
We say that the orbifold O will have the end fundamental group condition if the
fundamental group of each of its end has the unique fixed direction property for all
13
representations in
HomE (π1 (O), Aff(Rn ))
that arises as holonomy homomorphisms of affine structures on O with radial ends.
Finally, we say that the orbifold O will have the convex end fundamental group
condition if the fundamental group of each of its end has the unique fixed direction
property for all representations in HomE (π1 (O), Aff(Rn )) that arises as holonomy homomorphisms of properly convex affine structures on O with radial ends.
For example, if each end of O has singularity of dimension 1, then this is true: If O
is affine with parallel end, then the singularity line in the universal cover of O is in the
parallel direction and determines the eigendirection.
If O is a 3-or 4-dimensional radiant affine manifold with parallel ends and has two
dimensional singularities in the ends, then in the universal cover the singularities are in
the planes containing the origin and the planes both contain lines in parallel direction
and hence they must meet in a line, which has the unique eigendirection.
Given a real projective orbifold O with our conditions, we define a subspace of
HomE (π1 (O), PGL(n + 1, R)) to be the subspace where for each end E the holonomy
restrict in π1 (E) to one fixing a common point in RP n . Suppose now that O is a real
projective orbifold and V is an open subset of
HomE (π1 (O), PGL(n + 1, R))
invariant under the conjugation action so that one can choose a continuous section
sV : V → (RP n )e sending a holonomy to a common fixed point of π1 (E) for the
choice of π1 (E) as a subgroup of π1 (O) and sV satisfies sV (gh(·)g −1 ) = gsV (h(·)) for
g ∈ PGL(n + 1, R)). There might be more than one choice of section in certain cases.
Again, e is the number of ends of O and sV is said to be a fixed-point section.
We say that V and π1 (E) have the unique fixed direction property if the holonomy
of the radial end E has a common fixed point of, then it is the point unique by
some unspecified conditions fixed for U. (Of course, the condition could be an empty
condition.) The subset V and the orbifold O will have the end fundamental group
condition if the fundamental group of each of its end for every holonomy homomorphism
in V has the unique fixed point property.
Finally we say that the orbifold O will have the end fundamental group condition if
the fundamental group of each of its end has the unique fixed point property for all
representations in HomE (π1 (O), PGL(n+1, R)) that arise as holonomy homomorphisms
of real projective structures on O. We say that the orbifold O will have the convex
end fundamental group condition if the fundamental group of each of its end has the
unique fixed point property for all representations in
HomE (π1 (O), PGL(n + 1, R))
that arise as holonomy homomorphisms of real projective structures on O and acting
on properly convex subsets.
Actually, it is not a purely group condition but a geometric condition. In fact, it
might be possible that such a condition holds for a subset of representation space but
not for some other subsets of HomE (π1 (O), PGL(n + 1, R)). In such cases, our results
are valid for the components where the conditions hold.
14
If O is real projective and has a singularities of dimension one, then the universal
cover of O has more than two lines corresponding to singular loci. The developing image
of the lines must meet at a point in RP n , which is a fixed point of the holonomy group
of an end. If O has dimension 3, this means that the end surface has corner-reflectors
or cone-points.
Also, let O be a real projective 3-orbifold with radial ends. If each end has Euler
characteristic < 0, then the end fundamental group condition holds: Each end E has a
2-dimensional real projective structure. Take a surface cover and the cover has a real
projective structure. By the convex decomposition theorem, the surface decomposes
into convex subsurfaces and π-annuli. The convex subsurface is strictly convex. An
end neighborhood has a universal cover that decomposes into a neighborhood of the
cone over the universal cover of convex subsurfaces and annuli. The cone vertex of the
universal cover of the convex subsurface is unique fixed point of the deck transformation
group acting on it. If there are more than two fixed ponts for the holonomy of a surface,
then then the set of hyperspaces through the two points is an invariant set and hence
we obtain a geodesic foliation on the end. Hence, there is a unique fixed point.
If O is a real projective n-orbifolds with radial and the hyperbolic end fundamental
groups then convex end fundamental group condition holds. This follows since the
end orbifold has strictly convex real projective structures with irreducible holonomy
homomorphism as follows from Benoist’s work [5].
3.2. Local homeomorphism theorems. We denote by repE (π1 (O), Aff(Rn )) to be
the subspace quotient out by the action of conjugation h(·) → gh(·)g −1 for g ∈ Aff(Rn ).
We denote by repE (π1 (O), PGL(n + 1, R)) to be the subspace quotient out by conjugation by elements of PGL(n+1, R). This involves a choice of π1 (U ) in π1 (O) as one can
compose with a path in O. Let V be the open subspace of HomE (π1 (O), PGL(n+1, R))
invariant under the conjugation action so that one can choose a continuous eigenvectorsection sV : V → (RP n )e sending a holonomy to the fixed point of π1 (U ) for the choice
of π1 (U ) as a subgroup of π1 (O). Note that sV satisfies sV (gh(·)g −1 ) = gsV (h(·)) for
g ∈ PGL(n + 1, R)). As above, there might be more than one choice of section in
certain cases.
We mention that often V might be the entire representation space or a component of
it and sV can be defined by a natural choice and even uniquely if the end fundamental
group conditions are satisfied.
We define the end restricted deformation space Def A,E (O) on O to be the subspace of
the deformation space Def A (O) of affine structures on O where each end is parallel. We
define Def A,E,U ,sU (O) to be the subspace of Def A,E with holonomy in the open subset
U of HomE (π1 (O), Aff(Rn )) invariant under the conjugation action and with affine
structures so that the end direction is given by sU where U is a conjugation-invariant
subset of Hom(π1 (O, Aff(Rn )). sU satisfies sU (gh(·)g −1 ) = gsU (h(·)) for g ∈ Aff(Rn ).
We also define the end restricted deformation space Def E (O) to be to be the subspace
of the deformation space Def(O) of real projective structures on O where each end is
radial. Again Def E,U ,sU (O) is defined to be the subspace with holonomy in U and the
end determined by sU , i.e., the geodesic foliations are concurrent to the fixed points as
given by sU .
15
The rest of the proof of the first part of Theorem 3.1 is similar to [18]. We cover O by
open sets covering O0 and open sets which are end-parallel. We order the end-parallel
open sets to be prior to the open sets covering O0 .
Theorem 3.1. Let O be a topologically tame n-orbifold with radial ends. Let U is a
conjugation-invariant open subset of HomE (π1 (O), Aff(Rn )), and U 0 be the image in
repE (π1 (O, Aff(Rn )). The map
hol : Def A,E,U ,sU (O) → repE (π1 (O), Aff(Rn ))
sending affine structures determined by the eigenvector-section sU to the conjugacy
classes of holonomy homomorphisms is a local homeomorphism on an open subset U 0
in repE (π1 (O), Aff(Rn )).
Theorem 3.2. Let O be a topologically tame n-orbifold with radial ends and V a
conjugation-invariant open subset of
Hom(π1 (O, PGL(n + 1, R)),
and V 0 the image in
rep(π1 (O, PGL(n + 1, R)).
Let sV be the fixed-point section defined on V with images in (RP n )e . Then the map
hol : Def E,V,sV (O) → repE (π1 (O), PGL(n + 1, R))
sending the real projective structures with ends compatible with sV to their conjugacy
classes of holonomy homomorphisms is a local homeomorphism restricted to an open
subset V 0 .
Note that the last part is proved by using affine suspension. This will prove Theorem
A by setting V to be the entire set as O has the end fundamental group condition.
If O has the end fundamental group conditions, then we set U and V to be the
corresponding character spaces.
Corollary 3.3. Suppose that O is a topologically tame n-orbifold with the end fundamental group conditions. The map hol : Def A,E (O) → repE (π1 (O), Aff(Rn )) sending
an affine structure to the conjugacy class of its holonomy is a local homeomorphism.
So is the map hol : Def E (O) → repE (π1 (O), PGL(n + 1, R)).
3.3. The proof of the local homeomorphism theorems. We wish to now prove
Theorem 3.1 following the proof of Theorem 1 in Section 5 of [18].
Let O be an affine orbifold with the universal covering orbifold Õ with the covering
map p : Õ → O and let the fundamental group π1 (O) act on it as an automorphism
group.
Let U and sU be as above. We will now define a map hol : Def A,E,U ,sU (O) →
repE (π1 (O), Aff(Rn )) by sending the affine structure to the pair (dev, h) and to the
conjugacy class of h finally. The continuity of hol is easy to show in fact for any
geometric structures. We cover O by open sets as in [18]. There is a codimension-0
compact submanifold O0 of O so that π1 (O0 ) → π1 (O) is surjective. The holonomy
is determined on O0 . Since the deformation space has C r , r ≥ 1, topology induced
by dev, it follows that small change of dev in C r -topology implies sufficiently small
16
change in h(gi0 ) for generators gi0 of π1 (O0 ) and hence sufficiently small change of h(gi )
for generators gi of π1 (O). (Actually for the continuity, we do not need any condition
on ends.)
For the purpose of this paper, we use r ≥ 2. We will use this fact a number of times.
By a v-parallel set, we mean a subset of Rn which is invariant under the translation
along positive multiples of a fixed nonzero vector v. That is, it should be a union of
the images under translations along positive multiples of a nonzero vector.
An end-parallel subset of Õ or Rn is a v-parallel set where v is the eigenvector of the
linear parts of the corresponding end detemined by sU .
3.3.1. Affine structures. To show local homeomorphism property, we take an affine
structure (dev, h) on O and the associated holonomy map h. We cover Õ0 by small
precompact open sets as in Section 5 of [18]. We cover O − O0 by long open sets in
the parallel direction. Consider Lemmas 3, 4, and 5 in [18]. We can generalize these
to include v-parallel sets for invariant direction v of the finite group GB where v is the
eigenvector of eigenvalue 1 since GB is finite. We repeat them below. The proofs are
very similar and use the commutativity of translation by eigenvector with the action
of GB .
Lemma 3.4. Let GB be a finite subgroup of Aff(Rn ) acting on v0 -parallel open subset
B of Rn for eigenvector v0 of the linear part of GB . Let ht : GB → G, t ∈ [0, ), > 0, be
an analytic parameter of representations of GB so that h0 is the inclusion map. Let vt
is a nonzero eigenvector of ht (GB ) for each t and we assume that t 7→ vt is continuous.
Then for 0 ≤ t ≤ , there exists a continuous family of diffeomorphisms ft : B → Bt to
vt -parallel open set Bt in X so that ft conjugates h(GB )-action to ht (GB )-action; i.e.,
ft ht (g)ft−1 = h0 (g) for each g ∈ GB and t ∈ [0, ].
Here vt , vh0 and vh0 ,t below are of course determined by sU .
Lemma 3.5. Let GB be a finite subgroup of Aff(Rn ) acting on v0 -parallel open subset
B of Rn for eigenvector v0 of the linear part of GB . Suppose that h is a point of an
algebraic set V ⊂ Hom(GB , Aff(Rn ) for a finite group, and let C be a cone neighborhood
of h. Suppose that h is an inclusion map. Suppose that vh0 is the eigenvector of the
linear part of h0 (GB ) for each h0 ∈ C and the map h0 7→ vh0 is a continuous map
C → Rn . Then for each h0 ∈ C, there is a corresponding diffeomorphism fh0 : B →
Bh0 , Bh0 = fh0 (B) so that fh0 conjugates the h(GB )-action on B to the h0 (GB )-action
0
on Bh0 ; i.e., fh−1
0 h (g)fh0 = h(g) for each g ∈ GB where Gh0 is a vh0 -parallel open set.
Moreover, the map h0 7→ fh0 is continuous from C to the space C ∞ (B, X) of smooth
functions from B to X.
Continuing to use the notation of Lemma 3.5, we define a parameterization l :
S × [0, ] → C which is injective except at S × {0} which maps to h. (We fix l although
C may become smaller and smaller). For h0 ∈ S, we denote by l(h0 ) : [0, ] → C
be a ray in C so that l(h0 )(0) = h and l(h0 )() = h0 . Let the finite group GB act
on a vh -parallel submanifold F of vh -parallel open set B for eigenvector vh of h(GB ).
Let vh0 ,t be a nonzero eigenvector of l(h0 )(t) for h0 ∈ S and t ∈ [0, ] and we suppose
that S × [0, ] → Rn given by (h0 , t) → vh0 ,t is continuous. A GB -equivariant isotopy
17
H : F × [0, ] → Rn is a map so that Ht is an imbedding for each t ∈ [0, 0 ], with
0 < 0 ≤ , conjugating the GB -action on F to the l(h0 )(t)(GB )-action on Rn , where
H0 is an inclusion map F → Rn where the image H(F, t) is vh0 ,t -parallel set for each
t. The above Lemma 3.5 says that for each h0 ∈ C, there exists a GB -equivariant
isotopy H : B × [0, ] → Rn so that the image H(B, t) is vh0 ,t -parallel open set for each
t. We will denote by Hh0 ,0 : B → Rn the map obtained from H for h0 and t = 0 .
Note also by the similar proof, for each h0 ∈ S, there exists a GB -equivariant isotopy
H : F × [0, 00 ] → Rn .
Lemma 3.6. Assume the above paragraph. Let H : F × [0, ] × S → Rn be a map so
that H(h0 ) : F × [0, 0 ] × S → Rn is a GB -equivariant isotopy of F for each h0 ∈ S
where 0 < 0 ≤ for some > 0. Then H can be extended to Ĥ : B × [0, 00 ] × S → Rn
so that Ĥ(h0 ) : B × [0, 00 ] → Rn is a GB -equivariant isotopy of B for each h0 ∈ S where
0 < 00 ≤ . The image Ĥ(h0 )(t)(B) is a vh0 ,t -parallel open set for each h0 , t.
We define the inverse map from a neighborhood in rep(π1 (O), Aff(Rn )) of the image
point. This is accomplished as in [18] for precompact open covering sets and for long
open covering sets we use the above lemmas. This is so since we are working with
finitely many open sets.
Also, finally, we need to prove local injectivity of hol as in the last step of the
proof of Theorem 3.1. Given two structures in a sufficiently small neighborhood of the
deformation space, we show that if their holonomy homomorphisms are the same, then
we can isotopy one to the other using vector fields as in [18]. Here, our orbifolds are not
necessarily compact but has Riemannian metric at its end that is flow-invariant. These
vector fields will be uniformly C 1 -bounded by a small uniform constant depending on
the two structures. Hence, it is sufficient to estimate differences of maps in a compact
submanifolds and the estimates will propagate to the end. This completes the proof of
Theorem 3.1.
3.3.2. Real projective structures. Suppose now that O is a real projective orbifold of
dimension n. We assume that O has an end fundamental group conditions. Then
let O0 be the affine suspension of O, which has a parallel end with end direction
determined by the radial ends of O. We define Def sA,E,U ,sU (O0 ) as the subspace of the
deformation space consisting of elements where whose structure is ones obtained from
affine suspension constructions. repsE (π1 (O0 ), Aff(Rn+1 )) is the subspace of elements
whose representations are so that the center of π1 (O0 ) = π1 (O) × Z always maps to a
dilatation. By Theorem 3.1, we have
hol : Def sA,E,U ,sU (O0 ) → repsE (π1 (O0 ), Aff(Rn+1 ))
is a local homeomorphism.
By the affine suspension construction, π1 (O0 ) is isomorphic to π1 (O) × Z, and the
generator of Z is always mapped to a dilatation, elements of π1 (O) fixe the fixed point
of the dilatation. We can choose the fixed point to be the origin by changing the
developing map by a post-composition with a translation.
We define repE (π1 (O), GL(n+1, R)) and repE (π1 (O), SL± (n+1, R)) as the respective
subsets where the the holonomy groups of the ends of O have common eigenvectors.
18
We note that repE (π1 (O0 ), Aff(Rn+1 )) is identical with repE (π1 (O), GL(n + 1, R)) ×
R+ which is the subspace of rep(π1 (O), GL(n + 1, R)) where the fundamental group
of each end has an eigendirection. repE (π1 (O), GL(n + 1, R)) can be identified with
repE (π1 (O), SL± (n + 1, R)) × H1 (π1 (O), R) by using map
GL(n + 1, R) → SL± (n + 1, R) × R+
which is given by sending a matrix L to (L/| det(L)|, log(| det(L)|)).
We define Def Sn (O) as the deformation space of (Sn , SL± (n + 1, R))-structures on
O. and Def Sn ,E (O) as the deformation space (Sn , SL± (n + 1, R))-structures on O
with radial ends. Given an open subset U in repE (π1 (O0 ), Aff(Rn+1 )), an eigenvector
section map sU induces a map q ◦ sU for the quotient map q : R(n+1)e − {O} → (Sn )e .
We assume that U is an open subset invariant under changing the holonomy of the
generators of Z to be any scalar in R+ . Moreover, U corresponds to an open subset U 0 of
HomE (π1 (O), SL± (n+1, R)) invariant under conjugation. We define Def Sn ,E,U 0 ,q◦sU 0 (O)
as the subspace of where holonomy in U 0 and the end compatible with q ◦ sU 0 , i.e., an
end neighborhood of each structure is foliated by concurrent geodesics determined by
q ◦ sU 0 .
Define H 1 (O, R) as the space of homomorphisms π1 (O) → R. A radiant affine
structure on O0 can also be studied using its projection to Sn . We can cover O0 by
radial cone with vertex at the origin and project to Sn . Each gluing of open radial cones
becomes an element of SL± (n+1, R) acting on Sn with positive scalar factors forgotten.
The positive scalar factor information can be put into an element of H 1 (O, R). From
this, we deduce that Def sA,E,U ,sU (O0 ) is homemorphic to Def Sn ,E,U 0 ,q◦sU 0 (O)×H 1 (O, R)×
R+ − {1}:
We see that hol restricts to the inclusion map for the last factor R+ − {1} → R+
and restricts to identity on H 1 (O, R). As a consequence
Def Sn ,E,U 0 ,q◦sU 0 (O) → repE (π1 (O), SL± (n + 1, R))
is a local homeomorphism.
The quotient map SL± (n + 1, R) → PGL(n + 1, R) is induced by the action of the
group {I, −I} and is a double covering map. The induced map
q̂ : repE (π1 (O), SL± (n + 1, R)) → repE (π1 (O), PGL(n + 1, R))
is also induced by the same group action.
The map q̃ : Def Sn ,E (O) → Def E (O) is also induced by action (dev0 , h0 ) → (q ◦
dev0 , q̂(h0 )). We have a commutative diagram:
q̃
Def Sn ,E (O) → Def E (O)
↓ hol
↓ hol
(1)
q̂
repE (π1 (O), SL± (n + 1, R)) → repE (π1 (O), PGL(n + 1, R))
There is a section s̃ : Def E (O) → Def Sn ,E (O) given as follows. Given a real projective
structure on O, there is a development pair (dev, h) where dev : Õ → RP n is an
immersion and h : π1 (O) → PGL(n + 1, R) is a homomorphism. Since Sn → RP n is a
covering map and Õ is a simply connected manifold while O being a good orbifold, there
19
exists a lift dev0 : Õ → Sn unique up to the action of {I, −I}. This induces an oriented
projective structure on Õ and dev0 is a developing map for this geometric structure.
Given a deck transformation γ : Õ → Õ, we have that dev0 ◦γ is again a developing map
for the geometric structure and hence equals h0 (γ) ◦ dev0 for h0 (γ) ∈ SL± (n + 1, R). We
verify that h0 : π1 (O) → SL± (n + 1, R) is a homomorphism. This means that (dev0 , h0 )
gives us an oriented real projective structure which induces the original real projective
structure.
The map s̃ is clear continuous since the C 1 -closeness of the developing map to RP n
means the C 1 -closeness of the lifts. Thus, we showed that
Theorem 3.7. q̃ : Def Sn ,E (O) → Def E (O) is a homeomorphism.
From the exact sequence
{−I, I} → SL± (n + 1, R) → PGL(n + 1, R),
we can easily show that there is an exact sequence:
(2)
q̂ 0
B1 := Hom(π1 (O), {−I, I}) → HomE (π1 (O), SL± (n+1, R)) → HomE (π1 (O), PGL(n+1, R)).
Now taking quotient by SL± (n + 1, R) and PGL(n + 1, R) respectively, we obtain the
exactness of:
(3)
q̂
Hom(π1 (O), {−I, I}) → repE (π1 (O), SL± (n + 1, R)) → repE (π1 (O), PGL(n + 1, R)).
Since the first group B1 is finite, q̂ is a branch covering map branch locus that is a
fixed point of some subgroup of B1 consisting of order two elements.
There is a local section defined on the local neighborhoods of the image U of hol in
repE (π1 (O), SL± (n + 1, R)) :
Let U 0 be the inverse image of U in
HomE (π1 (O), PGL(n + 1, R)).
Then there is a local section from local neighborhoods in U 0 to
HomE (π1 (O), SL± (n + 1, R))
since from a real projective structure one can always construct an (Sn , SL± (n + 1, R))structure. It follows that B1 acts locally freely on the image of hol in
repE (π1 (O), SL± (n + 1, R))
so that we can conclude that q̂|U 0 is a local homeomorphism. This proves Theorem 3.2
by equation 1.
In general, there is a following theorem used commonly but not written anywhere.
Theorem 3.7 can be refined so that the holonomy lies in SL(n + 1, R) and the lifted
structure is a (Sn , SL(n + 1, R))-structure.
Corollary 3.8. Let M be an orientable orbifold possibly closed or open with radial
ends. Suppose that h : π1 (M ) → PGL(n + 1, R) is a holonomy of real projective
structure with radial ends. Then we can lift to h0 : π1 (M ) → SL(n + 1, R). Moreover,
20
it is a holonomy of (Sn , SL± (n + 1, R))-structure lifting the real projective structure. So
the image of the hol in rep(π1 (M ), PGL(n + 1, R)) is homeomorphic to that of hol in
rep(π1 (M ), SL(n + 1, R)) whenever M is an orientable orbifold.
Proof. Recall SL(n + 1, R) is the group of orientation-preserving linear automorphisms
of Rn+1 and hence is the group of orientation-preserving automorphisms of Sn . Since
the deck transformations of the universal cover M̃ of the lifted (Sn , SL± (n + 1, R))orbifold are orientation-preserving, it follows that the holonomy of the lift are in SL (n+
1 , R). We use as h0 the holonomy of the lifted structure.
4. Convexity
4.1. Convexity and convex domains. An affine manifold is convex if every path
can be homotoped to an affine geodesic with endpoints fixed. A complete real line in
RP n is a 1-dimensional subspace of Rn with denote it by R. An affine manifold is
properly convex if there is no affine map from R into it, i.e., there is no complete affine
line in its universal cover.
Proposition 4.1. (Vey) An affine manifold with nonempty parallel end is convex if
and only if the developing map sends the universal cover to a convex open domain in
Rn . An affine manifold with nonempty parallel end is properly convex if and only if the
developing map sends the universal cover to a properly convex open domain in Rn .
Proof. The first part is Theorem 8.1 of Shima [50]. The second part is Theorem 8.3
of [50] since the hyperbolicity there is equivalent to proper convexity. (See Kobayashi
[41].)
Lemma 4.2. A properly convex subset of an affine patch is a convex subset of a compact
subset of possibly another affine patch.
Proof. This is similar to the 2-dimensional situation in Section 1.3 of [16].
n
n
A complete real line in RP is a 1-dimensional subspace of RP with one point
removed with denote it by R. That is, it is the intersection of a 1-dimensional subspace
by an affine patch. A convex projective geodesic is a projective geodesic in a real
projective manifold which lifts to a projective geodesic the image of whose composition
with a developing map do not contain a complete real line. A real projective manifold is
convex if every path can be homotoped to a convex projective geodesic with endpoints
fixed. It is properly convex if there is no projective map from the complete real line R.
Proposition 4.3. (Vey)
• A real projective orbifold with nonempty radial end is convex if and only if the
developing map sends the universal cover to a convex open domain in RP n .
• A real projective orbifold with nonempty radial end is properly convex if and
only if the developing map sends the universal cover to a properly convex open
domain in a compact domain in an affine patch of RP n .
• If a convex real projective orbifold with nonempty radial end is not properly
convex, then its holonomy is reducible.
21
Proof. The first part follows by affine suspension and Proposition 4.1. For the second
part, the affine suspension has a developing image to a properly convex subset of Rn
an affine patch and Lemma 4.2. For the final item, a convex subset of RP n is a convex
subset of an affine patch An , isomorphic to an affine space. A convex subset of An
that has a complete affine line must contain a maximal complete affine subspace. Two
such complete maximal affine subspace do not intersect since otherwise there is a larger
complete affine subspace of higher dimension by convexity. We showed in [14] that the
maximal complete affine subspaces are parallel. This implies that the boundary of the
affine subspaces are lower dimensional subspaces. These subspaces are preserved under
the group action.
See [8] for the proofs of the following propositions.
Proposition 4.4. (Benoist) Suppose that a discrete subgroup Γ of PGL(n, R) acts on
a properly convex (n − 1)-dimensional open domain Ω so that Ω/Γ is compact. Then
the following statements are equivalent.
• Every subgroup of finite index of Γ has a finite center.
• Every subgroup of finite index of Γ has a trivial center.
• Every subgroup of finite index of Γ is irreducible in PGL(n, R). That is, Γ is
strongly irreducible.
• The Zariski closure of Γ is semisimple.
• Γ does not contain a normal infinite nilpotent subgroup.
• Γ does not contain a normal infinite abelian subgroup.
The group with property (i) above is said to be the group with trivial virtual center.
Theorem 4.5. (Benoist) Let Γ be a discrete subgroup of PGL(n, R) with a trivial virtual center. Suppose that a discrete subgroup Γ of PGL(n, R) acts on a properly convex
(n − 1)-dimensional open domain Ω so that Ω/Γ is compact. Then every representation
of a component of Hom(Γ, PGL(n, R)) containing the inclusion representation also acts
on a properly convex (n − 1)-dimensional open domain cocompactly.
We call the group such as above theorem the vcf-group. By above Proposition 4.4,
we see that every representation of the group acts irreducibly.
Recall that the Hausdorff distance between two convex subsets K1 , K2 of RP n is
defined by
dH (Cl(K1 ), Cl(K2 )) < if and only if Cl(K1 ) ⊂ N (Cl(K2 )), Cl(K2 ) ⊂ N (Cl(K1 ))
where N (A) is the -neighborhood of A under the standard elliptic metric of RP n for
> 0.
Corollary 4.6. Suppose that the fundamental group Γ of a closed (n−1)-orbifold Σ is a
virtual product to hyperbolic groups and abelian groups. Given a parameter µt , t ∈ [0, 1]
of convex RP n−1 -structures on Σ, we can find a continuous family of developing maps
Dt and holonomy homomorphism ht : Γ → Γt so that Kt := Cl(Dt (Σ̃)) is a uniformly
continuous family of convex domains in RP n−1 under the Hausdorff metric of the closed
subsets of RP n−1 .
22
Proof. First, we assume that Γ is a hyperbolic group. Since
hol : Def(Σ) → rep(Γ, PGL(n, R))/PGL(n, R)
is a homeomorphism, we can choose Dt and ht as above.
We will show that given 0 < < 1/2 and t0 , t1 ∈ [0, 1], we can find δ > 0 such if
|t0 − t1 | < δ, then Kt1 ⊂ N (Kt0 ) and Kt0 ⊂ N (Kt1 ).
To show this, fix t0 and choose a finite number of generating elements ht0 (g1 ), ..., ht0 (gn )
of Γ so that /2-neighborhoods of their fixed points cover ∂Kt0 by [6]. (They all have
unique pairs of attracting and repelling fixed points (at0 ,i , rt0 ,i ) and unique invariant
supporting hyperspaces Ha,t0 ,i , Hr,t0 ,i at the fixed points.) Furthermore, we choose so
that {Ha,t0 ,i , Hr,t0 ,i |i = 1, .., n} is /8-dense in the set of supporting hemispheres of Kt
using duality. Here, the intersection of Ha,t0 ,i and Hr,t0 ,i is a unique ht0 (gi )-invariant
convex domain Bt0 ,i containing Kt0 . Then by geometry, and estimating with derivatives, we obtain
n
\
Kt0 ⊂
Bt0 ,i ⊂ N/4 (Kt0 )
i=0
T
In fact, for sufficiently small > 0, any two faces ni=0 Bt0 ,i with points of distance ≤ are /8-close in the Hausdorff sense which follows since ∂Kt is C 1 .
Notice that
at,i , rt,i , Ha,t,i , Hr,t,i and Bt,i
depends only on ht (gi ) in continuous manners in the spherical metrics and the Hausdorff
metrics. It follows that for a sufficiently small δ > 0
Kt ⊂
n
\
Bt,i ⊂ N/2 (Kt ) for |t − t0 | < δ.
i=0
T
It also follows that for a sufficiently small δ > 0, any two faces of ni=0 Bt,i with points
of distance ≤ are /4-close in the Hausdorff sense.
We choose δ > 0 so that for |t − t0 | < δ, the fixed points at,i , rt,i of ht (gi ) and
the invariant hyperspaces Ha,t,i and Hr,t,i move from the ones at t0 by a distance less
than /4 and the Hausdorff distance between Bt,iTand Bt0 ,i is less than /4.
Tn From this
n
and estimating with hyperspaces, we can show i=0 N/4 (Bt,i ) ⊂ N/2 ( i=0T
Bt,i ) for
|t − t0 | < δ and sufficiently small > 0 as any point has the closest point in ni=0 Bt,i
at a point in the interior of a face or the intersection of at least two faces. From this,
it follows that
n
\
i=0
Bt0 ,i ⊂
n
\
N/4 (Bt0 ,i ) ⊂
i=0
n
\
N/2 (Bt,i ) ⊂ N/2 (
i=0
n
\
i=0
for |t − t0 | < δ and sufficiently small > 0. Conversely, we have
n
\
Bt,i ⊂ N/2 (
i=0
n
\
i=0
23
Bt0 ,i ).
Bt,i )
Since we also have
Kt ⊂
n
\
Bt,i ⊂ N/2 (Kt ),
i=0
we obtain
Kt0 ⊂
n
\
Bt0 ,i ⊂ N/2 (
i=0
n
\
Bt,i ) ⊂ N (Kt )
i=0
and vice versa.
If Γ is a nontrivial product, then we can choose a parameter of invariant convex
domains for each factor groups and their joins should satisfy the conclusion.
4.2. Definitions associated with ends.
Lemma 4.7. Suppose that O is a topologically tame properly convex real projective
orbifold with radial ends and a universal cover Õ. Let Ũ be inverse image of the union
U of mutually disjoint end neighborhoods. For a given component U1 of Ũ , we have
if γ(U1 ) ∩ U1 6= ∅, then γ(U1 ) = U1 and γ lies in the fundamental group of the end
associated with U1 . Also, if γ(U1 ) ∩ κ(U1 ) 6= ∅, then γ −1 κ(U1 ) = U1 and γ −1 κ lies in
the fundamental group of the end.
Proof. This follow since U1 covers an end neighborhood.
Proposition 4.8. Suppose that O is a topologically tame properly convex real projective
orbifold with radial ends, and its developing map sends its universal cover Õ to a convex
domain. Let U and Ũ be as above. Then the closure of each component of Ũ contains
the endpoints p of the leaf of radial foliation in Ũ lifted from U , and there exists a unique
one for each component U1 of Ũ and the subgroup of h(π1 (O)) fixing it is precisely the
image of the end fundamental group corresponding to U1 .
Definition 4.9. We will simply identify Õ with the developing image that is the convex
domain in RP n . We call such a vertex p the end vertex of Õ and we associate the unique
component U1 of Ũ , a so-called end neighborhood of the vertex p and associate a unique
subgroup, to be denoted by Γp of π1 (O) corresponding to U1 , so-called end fundamental
group corresponding to the vertex p or the component U1 . We generalize the notion of
end neighborhood as an Γp -invariant open set containing a component of Ũ for U as
above.
We can associate an end E of O with any end vertex p. Thus, there are only finitely
many orbit types for end vertices. Sometimes, we denote Γp by π1 (E) also and is said
to be the end fundamental group. Given an end E corresponding to p, we denote by
Rp (E) the space of rays from p in Õ. Also, for a subset K of U1 , we denote by Rp (K)
the space of rays from p ending at K. We denote by RPpn−1 and Rnp the projective
space and the vector space associated with p at the developing image of Õ, where
Rp (E), Rp (K) ⊂ RPpn−1 .
Proposition 4.10. If O is convex, Rp (E) is also a convex domain in an affine patch
of RP n−1 for the associated vertex p for E. The group h(π1 (O)) induces a group
24
h0 (Γ(p)) of projective transformation of RP n−1 acting on Rp (E) and Rp (E)/h0 (Γ(p))
is diffeomorphic to the end orbifold E and has the induced projective structure of E.
Proof. Straightforward.
Remark 4.11. From now on, by an end of an orbifold E, we will mean the end in the
orbifold or the corresponding object for the universal cover Õ and we will use π1 (E)
the subgroup of π1 (O) fixing this object. We see that each end E of the universal
cover corresponds in a one-to-one manner to a unique end vertex v and vice versa fixed
by the holonomy group of the end so that γ(v) = π1 (E). Also, Ẽ will denote the set
Rp (E).
Remark 4.12. From now on, our orbifold O will be convex, and we can identify Õ with
the developing image and h(π1 (O) will act on it. We will actually identify π1 (O) with
h(π1 (O) as well.
4.3. Properties of ends.
4.3.1. Properties of horospherical ends.
Proposition 4.13. Let O be a topologically tame properly convex real projective norbifold with radial ends.
• For each horospherical end, the space of ray from the end point form a complete
affine space of dimension n − 1.
• The only eigenvalues of g for an element of a horospherical end fundamental
group are 1 or complex numbers of absolute value 1.
• An end point of a horospherical end cannot be on a segment in bdÕ.
• For any compact set K inside a horospherical neighborhood, there exists a horospherical ellipsoid neighborhood disjoint from K.
• Let E be a complete end. Suppose that π1 (E) has holonomy with eigenvalues of
absolute value 1 only. Then E is horospherical.
Proof. Let U be a hororspherical end with an end vertex p. The space of rays from
the end vertex form a convex subset ΩE of a complete affine space Rn−1 ⊂ RP n−1 and
covers an end orbifold OE with the discrete group π1 (E) acting as a discrete subgroup
Γ0E of the projective automorphisms so that ΩE /Γ0E is projectively isomorphic to OE .
From the paper [14], we see that ΩE is foliated by complete affine spaces of dimension
i with common boundary sphere of dimension i − 1 and the spaces of the leaves form
a properly open convex subset K of a projective space of dimension n − 1 − i. Then
Γ acts on K cocompactly but perhaps not discretely. Here, we aim to show i = 0 and
K is a point. Suppose that i ≥ 1. This implies that ΩE is a complete affine space of
dimension n − 1.
For each element g of Γ, a complex or negative eigenvalue of g cannot have a maximal
or minimal absolute value different from 1 since otherwise by taking the convex hull
of {g n (x)|n ∈ Z} for a point of U , we see that U must be not properly convex. Thus,
some of the largest and the smallest absolute value eigenvalues of g are positive.
25
Since g acts on a properly convex subset K of dimension ≥ 1, it follows that there
exists an eigenvalue > 1 and an eigenvalue < 1. Therefore, let λ1 > 1 be the greatest
absolute value eigenvalue and λ2 < 1 be the smallest. Let λ0 > 0 be the eigenvalue
associated with p. The possibilities are as follows
λ1 = λ0 > λ2 , λ1 > λ0 > λ2 , λ1 > λ2 = λ0 .
In all cases, there exists a point x∞ with largest or smallest positive eigenvalue distinct
from p. Again, x∞ ∈ Cl(U ) since x is a limit of g i (x) for i → ∞ or i → −∞. Since
x∞ 6∈ U , it follows that x = p. This is a contradiction. This implies that i = 0 and ΩE
is a complete affine space.
The second statement is clear from the above.
If p ∈ s for a segment s in bdΩ. Let us choose a maximal two-dimensional disk
D containing s with Do ⊂ Ωo . Then D ∩ ∂U contains an arc α so that α converges
to p in one direction. Let pi ∈ α be a point converging to p. There exists a deck
transformations gi in the end fundamental group at p so that gi (pi ) is in a fixed compact
fundamental domain F of ∂U . The gi fixes the end vertex p with eigenvalue 1 and gi
decomposes Rn+1 into blocks of matrix with diagonal entries 1 and the entries above
it all 1 and other entries zero and blocks of identity matrix and blocks of complex
eigenvalues of absolute values 1. There is a subspace P where gi restrict to the union
of blocks of identity and complex unit eigenvalues. Let ri be a sequence of bounded
projective automorphisms fixing p and ri ◦ gi (D) is in the plane containing D and
ri ◦ gi (l) is in the line containing l. Notice that α is differentiable at v since it is stricly
convex. Since the holonomy of gi |Do and gi |α are unbounded and α is tangent to l,
gi |l is unbounded and so is ri ◦ gi |l.
If ri ◦ gi (l) is bounded with length in between and π − for > 0, then ri ◦ gi has a
fixed point in the line containing l in a distance between and π − with eigenvalues
→ ∞. As ri is bounded, gi has eigenvalues → ∞ as i → ∞. Since the eigenvalues
of gi are always 1, this is a contradiction. Therefore, a subsequence of the lengths of
ri ◦ gi (l) goes to zero or to π. In either case, the lengths of gi (l) or gi−1 (l) converges to
π. This contradicts the proper convexity of Õ.
Choose an exiting sequence of end neighborhoods Ui and we take convex hulls Vi .
Vi is a union of n-simplicies with vertices in Ui . The sequence of the sets of vertices
are exiting. Therefore, Vi is also exiting. One can deform the boundary of Vi by small
amounts to make them strictly convex as there are no infinite straight lines in ∂Vi . We
can make Vi ellipsoid by taking even smaller ellipsoid neighborhood inside.
For the first item, suppose that E is not horospherical. Then there is a convex end
neighborhood U with the closure Cl(U ) so that Cl(U ) ∩ ∂ Õ has more than one points.
Suppose that bdU ∩ ∂ Õ is disjoint from the end vertex v. Since E is complete, then
we obtain that a small neighborhood N of a supporting hyperspace P at v is in ∂ Õ.
Since P is π1 (E)-invariant, we can act by π1 (E) to obtain a segment of dS -length π in
Cl(Õ) as a limit of a subsequence of the sequence gi (s) for a segment s in N containing
v in the interior so unless π1 (E) acts as identity on P . If π1 (E) acts as identity in P ,
then every complete ray through v is invariant considering the matrix expression and
we can easily obtain a segment of dS -length π in Cl(Õ).
26
Let F be a compact fundamental domain of Σ := bdU ∩ ∂ Õ under π1 (E).
By above, v is in the closure of Σ. Suppose that there exists a segment s in bdCl(Õ)
with distinct end points v and v 0 in Cl(Σ). Let P be a 2-plane containing s and meeting
Õ. Then P meets Σ in a convex arc α with end points v and v 0 . Moreover P contains a
half-plane H with v ∈ ∂H where ∂H is the unique supporting line in P to P ∩ Õ. We
choose a sequence xi ∈ α converging to v and gi ∈ π1 (E) so that gi (xi ) is in F . Then
gi (α) form a sequence of curves passing F . By choosing a subsequence, we assume
that the sequence gi (α) converges to a convex curve α∞ on Σ. We also assume that
gi (P ) converges to a plane P∞ containing α∞ and gi (H) to the unique half-space H∞
in P∞ with ∂H∞ the unique supporting line at v. Let a choose a bounded sequence
ri ∈ PGL(n + 1, R) so that ri gi (H) = H∞ for all i. Moreover, we may assume ri → I.
Then ri gi (α) also converges to α∞ . Suppose that α∞ has two end points v and
00
v . We may even assume that the endpoints of ri gi (α) equals that of α∞ . Then
ki := ri gi gj−1 rj−1 fixes the two points and ki acts on H∞ and forms a bounded sequence
of actions on ∂H∞ : Otherwise, ki |∂H∞ is represented by a diagonal matrix with entries
→ ∞, 0 which implies that gi gj−1 must contract or expand an interval in gj (s) that is
a neighborhood of a point vj00 near v 00 disjoint from v by an as larger an amount locally
as possible for sufficiently large i and send it to an interval containing a point vi00 near
v 00 bounded away from v also. Since gi gj−1 has eigenvalues 1 only, this certainly cannot
happen.
We can write ki in terms of a fixed coordinates of H∞ with v becoming [1, 0, 0] and
v 00 being [0, 1, 0] and a point y ∈ α be [0, 0, 1]:


λi 0 u i
ki =  0 δi vi  .
0 0 i
We know that λi /δi form a bounded sequence and ki ([0, 0, 1]) converges to [0, 1, 0].
Since we have ki ([0, 0, 1]) = [ui , vi , i ], it follows that ui /vi , i /vi → 0. There is also
sequence of elements xi ∈ α → v so that ki (xi ) is in a bounded neighborhood of
a compact subset of α∞ . Writing xi = [1, 1i , 2i ] so that 1i , 2i → 0, we obtain that
ki (xi ) = [λi /vi + 2i ui /vi , (δi /vi )1i + 2i , (i /vi )2i ], which form a subsequence converging
to v = [1, 0, 0], a contradiction.
Therefore any limiting α∞ has a unique end point v. We know that the geodesic flow
on E is ergodic since π1 (E) is abelian and hence amenable and the geodesic flow has
invariant transverse measure by Hirsch and Thurston [38]. Therefore, given any point
of E and a direction, we find a sequence as above converging to the geodesic in the
direction. Thus, it follows that for a dense set of directions, the corresponding curve in
Σ has only one end points. This implies that Σ has only one end point v. We showed
that our end is horospherical.
4.3.2. The properties of lens-shaped ends. We first need the following technical lemma
on recurrent arcs.
Lemma 4.14. Let O be a topologically tame properly convex real projective n-orbifold
with radial ends. Suppose gi ∈ PGL(n + 1, R) be a sequence of automorphisms so
27
that gi (x) = x for a lens-shaped end vertex x and l is a maximal segment in a lens.
(See Figure 2.) Let gi0 denote the induced projective automorphisms on RPxn−1 . gi0 (l0 )
converges geometrically to l0 where l0 is the projection of l to the projective link space
RPxn−1 of x. Furthermore, we suppose that
• l is in the disk D bounded by two segments s1 and s2 from x to and a convex
curve α with endpoints q1 and q2 in s1 and s2 respectively.
• β is another convex curve in D with endpoints in s1 and s2 so that α and β
and parts of s1 and s2 bound a convex disk in D.
• There is a sequence of points q̃i ∈ α converging to q1 and gi (q̃i ) ∈ F for a fixed
fundamental domain F of Õ.
• The sequences gi (D), gi (α), gi (β), gi (s1 ), and gi (s2 ) each converges to D, α, β, s1 ,
and s2 respectively.
Then we conclude that
• If the end points of α and β do not coincide at s1 or s2 , then α and β must be
straight geodesics from q2 .
• Suppose that the pairs of endpoints of α and β coincide and they are distinct
curves. Then there exists no segment in Cl(Õ) extending s1 or s2 properly.
Proof. There exists a bounded sequence of elements ri ∈ PGL(n + 1, R) so that
ri (gi (s1 )) = s1 and ri (gi (s2 )) = s2 and ri → I. Then ri ◦ gi is represented as an
element of PGL(3, R) in the projective plane containing D. Using x and endpoints of
s1 and s2 as standard basis,ri ◦gi is represented as a diagonal matrix. Moreover ri ◦gi (α)
is still converging to α as ri → I. Hence, this implies that the diagonal elements of
each ri ◦ gi are of form λi , µi , τi where λi → 0, τi → +∞ as i → ∞ and µi is associated
with x and τi is associated with q2 .
We have that ri ◦ gi (β) also converges to β. If the end point of β at s1 is different
from that of α, then log |λi /µi | forms a bounded sequence. In this case, β has to be a
geodesic from q2 and so is α. The similar argument hold for the s2 cases.
For the second item, µi /τi → 0, +∞ and λ/µi → 0, +∞ also since otherwise we can
show that β and α have to be geodesic with distinct endpoints. If there is a segment
s02 extending s2 , then ri ◦ gi (s02 ) converges to a great segment with endpoints x and −x
and so does gi (s02 ) as i → ∞ or i → −∞. This contradicts the proper convexity of O.
A concave end-neghborhood is an imbedded end neighborhood contained in a radial
end neighborhood in Õ that is a component of a complement of a lens-shaped domain
when the end is a lens-shaped. An open concave end set is the complement of a lens-part
in a lens-shaped cone.
The image is an end neighborhood up to taking a finite cover of O. If we take a
sufficiently large lens out, the image is an end neighborhood. In this case, its image
O is also said to be concave end-neghborhood of a corresponding end. (Note that the
second statement is a stronger condition.)
A trivial one-dimensional cone is a open half space in R1 given by x > 0 or x < 0.
28
s'
s'1
α
2
q
2
s2
β
s1
Figure 2. The figure for Lemma 4.14.
Recall that if π1 (E) is an admissible group, then π1 (E) has a finite index subgroup
in Zl × Γ1 × · · · × Γk for some l and k and Γi are hyperbolic. Here, we identify Õ as a
convex domain in RP n for convenience.
Let us consider E as a real projective (n − 1)-orbifold and consider Ẽ as a domain
in RP n−1 and h(π1 (E)) induces h0 : π1 (E) → PGL(n, R) acting on Ẽ.
Theorem 4.15. Let O be a topologically tame n-orbifold with radial ends. Let E be
a lens-shaped end of Õ. Assume that π1 (E) is an admissible group and is hyperbolic.
We denote by h0 (π1 (E)) the quotient image of h(π1 (E)) in PGL(n, R), the projective
automorphism group of the space RPVn of lines at an end vertex V of E.
(i) The complement of the manifold boundary of the lens-space domain D is a
nowhere dense set in bdCl(D) in RP n . Moreover, bdCl(D)−∂D is independent
of the choice of D. That is D is strictly lens-shaped.
(ii) The closure in RP n of a concave end-neighborhood of v contains every segment
l in bdÕ meeting the closure of a concave end neighborhood of v in lo . The
set S(v) of maximal segments from v in the closure of an end-neighborhood of
v is independent of the end-neighborhood and so is the union of S(v) and is
contained in the closure of any end neighborhood of v.
(iii) Any concave end neighborhood U of v under the covering map Õ → O, covers
the end neighborhood of E of form U/π1 (E).
(iv) S(g(v)) = g(S(v)) for g ∈ π1 (P) and S(v)o ∩ S(w) = ∅ or v = w for end
vertices v and w where S(v)o is the relative interior of S(v) in bdÕ.
Proof. (i) By Fait 2.12 [6], we obtain that π1 (E) is vcf and acts on irreducibly on a
proper convex cone and the cone has to be strictly convex by Theorem 1.1 of [4]. The
set of fixed points of h0 (π1 (E)) is dense [2].
We have a domain D with boundary components A and B transversal to the radial
lines. h(π1 (E)) acts on both A and B. It follows that the fixed points of h(π1 (E))
is dense in both the topological boundary A1 of A in Cl(A) and in the topological
boundary B1 of B in Cl(B).
We claim that for a nontrivial element g, the fixed points of h(g) in A1 are the same
as one one in B1 : There is a geodesic l fixed in Ẽ by h0 (g). This corresponds to a
29
2-dimensional subspace P , a projective plane, containing the cone-point v and fixed
points in A1 and ones in B1 . By the classification of projective trasformations on a
projective plane, h(g) can be represented by a diagonal matrix with positive entries.
Since A∩P and B ∩P are strictly convex curves acted upon by h(g), the transformation
h(g) can be represented by a diagonal matrix with positive entries λ, µ, τ with λµτ = 1
and λ > µ > τ and µ is associated with the end vertex v. This forces the two pairs of
fixed points in A1 and B1 to coincide. (By strict convexity, we see that the fixed points
are attracting and repelling ones.)
By density of the attracting and repelling fixed points A1 = B1 and A ∪ B is dense
in bdD.
Here, ∂D = A ∪ B. We see that bdCl(D) − ∂D is the closures of the attracting and
repelling fixed points of h(π1 (E)). Therefore this set is independent of the choice of D.
(ii) Consider any segment l in bdÕ meeting Cl(U1 ) for a concave end-neighborhood
U1 of v. This segment is contained in a union of segments from v. These segments are
all in the boundary of Cl(U1 ) by the fact that the segments of v can only end in the
interior of U1 or in the outer boundary of the lens or lies in the boundary of Cl(U1 ).
Thus, we may assume without the loss of generality that l be a segment from v in
Cl(U1 ) ∩ bdÕ. –(***)
If the interior of l contains a point p of bdCl(D) − A − B and is in the direction of an
end point of a recurrent geodesic m in Ẽ. Then m lifts to a convex arc m0 in A ending
at p. m0 and l is contained in a triangle T with vertex v and p and the other endpoint
of m0 . By recurrence, there exists a sequence gi ∈ π1 (E) so that gi (T ) approximates
T and ri ∈ PGL(n + 1, R) so that ri ◦ gi (T ) = T with p as a repelling fixed point and
ri → I. Then ri ◦ gi (l) converges to a great segment as above. This is a contradiction
as before by Lemma 4.14.
The set S(v) is clearly independent of the choice of the concave-end neighborhood
since it is the set of radial segments in the boundary of the concave end-neighborhood
that meets A1 only at the end.
(iii) Given a concave-end neighborhood N of an end vertex v, and any element
g ∈ π1 (E), we have that g(N ) = N for g ∈ Γ(v) or g(N ) ∩ N = ∅ otherwise: Suppose
that g(N ) ∩ N 6= ∅ and g(N ) 6= N : We also have g(N ) ∪ N 6= Õ since otherwise,
we have only two vertices while we must have infinitely many. Then, g(v) 6= v. Since
g(N ) ∩ N 6= ∅, and g(N ) ∪ N 6= Õ, we have a point x of bdg(N ) ∩ bdN ∩ O. By
concavity, there is a totally geodesic disk D containing x and included in N so that a
component N1 of Õ−D is in N . Similarly, there is a disk D0 containing x and contained
in g(N ) so that a component N10 of Õ−D0 is in N 0 . This implies that g(S(v)) = S(g(v))
must meet S(v). This means that there is a concave end neighborhood U1 of v meeting
g(U1 ) that covers a concave end neighborhood in O. Any element g of π1 (Õ) has the
property g(U1 ) = U1 or g(U1 ) ∩ U1 = ∅. Thus g(U1 ) = U1 and g ∈ Γ(v). Thus g(v) = v.
This is a contradictions.
(iv) Also, we have g(S(v)) = S(g(v)) for any end vertex v and the above implies
that S(v)o ∩ S(w) = ∅ or v = w for end vertices v and w.
30
Now we go to the cases when π1 (E) has more than two nontrivial factors abelian
or hyperbolic. The following theorem shows that lens-shaped ends are lens-shaped in
important cases.
Theorem 4.16. Let O be a topologically tame n-orbifold with radial ends. Suppose
that the holonomy of O is not virtually reducible and that π1 (E) has an infinite virtual
center. Let E be a lens-shaped end with the end vertex v. Then the following statements
hold:
(i) For RPvn−1 , we obtain
(i-1) Under h0 (π1 (E), Rn splits ino V1 ⊕· · ·⊕Vj and Ẽ is the quotient of the sum
C1 + · · · + Cl for properly convex or trivial one-dimensional cones Ci ⊂ Vi
for i = 1, . . . , l
(i-2) the Zarisky closure of h0 (π1 (E)) is the product G = G1 × · · · × Gl where Gi
is a reductive subgroup of GL(Vi ).
(i-3) Let Di denote the image of Ci in RPvn−1 . The number of hyperbolic group
factors of π1 (E) is ≤ l and the hyperbolic group factors of π1 (E) acts on
exactly one Di divisibly and other factors trivially.
(i-4) π1 (E) has a rank l free abelian group center.
(ii) The end is totally geodesic.
(iii) g ∈ Zl is diagonalizable with positive eigenvalues. For a nonidentity element
g ∈ Zl , the eigenvalue λ0 of g at x is strictly between its largest and smallest
eigenvalues.
(iv) The end is strictly lens-shaped and each Ci corresponds to a cone Ci∗ over a
totally geodesic (n−1)-dimensional domain Di0 corresponding to Di that contains
a concave open invariant set Ui . The end has a neighborhood that is a join of
D10 , .., Dl0 with v where the join of D10 , .., Dl0 forms the boundary.
(v) S(v) for nontrivial joined case is equal to the set of maximal segments with
S S
vertex v in ( ji=1 s∈Si (v) s) ∗ D10 ∗ · · · ∗ Ďi0 ∗ · · · ∗ Dl0 Si (v) is the set of maximal
segments with vertex v in Cl(Ci0 ) ∩ Ui .
Proof. (i) Without loss of generality by taking a finite index subgroup, we assume that
π1 (E) has a infinite free abelian center. Recall that E is an (n−1)-dimensional properly
convex projective orbifold. By Fait 1.5 in [6], Ẽ is a the radially projected image in
RPvn−1 of sum of convex cones C1 + C2 + ... + Cl and the abelian group Zl of rank
l acts as the centralizer subgroup in h0 (π1 (E)), and h0 (π1 (E)) acts on Ci irreducibly.
The center Zl acts trivially on each Ci by Benoist (See Proposition 4.4. in [4].)
Let Γ1 × · · · × Γk be the product of hyperbolic groups in π1 (E). Suppose that gi ∈ Γi
and gq ∈ Γq act on Cr nontrivially for i 6= j. Then they restrict to a semiproximal
automorphism by Benoist. [4] and gi and gj commute.
Let Gri and Grq denote the groups that are images of Γi and Γq in the automorphism
group of Cr . Since the action of h(π1 (E)) is irreducible on each Cr , the group Gr :=
⊕ki=1 Gri acts irreducibly on Cr with a compact Hausdorff quotient (Proposition 4.4. in
[4].) Suppose that more than two Gri are not trivial. Taking only nontrivial elements
gir ∈ Gri , they generate an abelian group.
31
Let Cr have dimension at least two. Let L(gir ) and M (gir ) denote the eigenspace of gir
associated with largest real and smallest real eigenvalues respectively. By positive semiproximality L(gir )∩Cl(Cr ) is a nonempty properly convex cone and so is M (gir )∩Cl(Cr ).
Then for q 6= i, gqr (L(gir )) is again an eigenspace of gir of largest real eigenvalue by
commutativity. Thus gqr (L(gir )) = L(gir ). Therefore, ⊕i6=j Gri acts on the convex hull
of (L(gir ) ∪ M (gir )) ∩ Cl(Ck ) and so does Gr . Then Cr has to equal this hull which
is actually a join of L(gir ) ∩ Cl(Cr ) and M (gir ) ∩ Cl(Cr ). This means that Gr acts
reducibly.
This is a contradiction and Gr = Gri for a unique i. Thus, Γi surjects onto Gr for
some pair of i and r.
Suppose that Γi nontrivially maps into the sum of Gr for all values of r where Γi
maps to a nontrivial group. Then this is surjective since otherwise there is an element
0
of some other Γp acting nontrivially on C k for some k 0 . But a hyperbolic group cannot
be a product group. It follows that Γi = Gr for exactly one r provided dim Cr ≥ 2 and
Γi acts trivially for all other r.
Hence, we showed that π1 (E) is virtually isomorphic to Zl × Γ1 × · · · Γl0 for l0 ≤ l.
where each Γi acts on the radially projected image of Cr properly and cocompactly for
exactly one r.
(ii) Let U1 be an end neighborhood of E in Õ Let S1 , ..., Sl be the projective subspaces
in general position meeting only at the end vertex v where factor groups Γ1 , ..., Γl act
irreducibly on. Let Ci0 denote the union of segments of length π from v corresponding
to the invariant cones in Si where Γi acts irreducibly for each i. The abelian center
Zl acts as identity on Ci in the projective space RPvn of lines through v. Let g ∈ Zl .
g|Cl0 is either identity or fixes a hyperspace Pi ⊂ Si not passing through v and g has a
representation as a scalar multiplication in the affine subspace Si − Pi of Si . Since g
commutes with every element of Γi acting on Ci0 , it follows that Γi acts on Pi as well.
Now consider Ci0 s so that for all g ∈ Zl acts as identity and take the join C and
consider those Ci0 s where some g ∈ Zl acts as nonidentity and let N be the indices of
such Ci0 s. Then every g ∈ Zl acts as identity on C and Ci0 ∩Pi for each i ∈ N . Moreover,
g is clearly diagonalizable with positive eigenvalues associated with the subspaces C or
Pi for i ∈ N .
There is a projection Π sending RPvn−1 to RP l given by sending each Ci to a point
ei and Zl acts naturally and cocompactly on ∆o with induced coefficients. Hence for
each Ci , we have an element g ∈ Zl so that g has the largest positive eigenvalue at Ci .
The limit set of g i (Õ) as i ∈ Z+ is a compact convex set Ci00 containing Ci0 ∩ Pi for
i ∈ N where Ci00 is really the image of a projection.
Since every element of Zl acts as identity on C, there is again an element g1 so that
g1 has the largest positive eigenvalue on C. The limit set of g1i (Õ) as i ∈ Z+ is a
compact convex set C 00 in C.
By the convexity and the Zl -invariance of Cl(Õ), we see that the join C 000 of C 00 and
all C”i for i ∈ N is in Cl(Õ). If there is a point z in Cl(Õ) − C 000 , then g1i (z) for
i ∈ Z+ limit to a point of C 00 . We can write a vector in Rn+1 for z as a positive linear
combinations of vector v 00 in C and vi00 in Pi for i ∈ N . If v 00 is not in C 00 , then g1i (z) for
i ∈ Z+ cannot limit to a point of C 00 as a matrix computation shows. Thus, v 00 ∈ C 00
32
and vi00 ∈ Ci0 ∩ Pi . Therefore, z ∈ C 000 . We obtain that C 000 = Cl(Õ), which implies that
h(π1 (O)) is virtually reducible, contradicting the premise.
Therefore, for all Ci , some g ∈ Zl acts as nonidentity. Then the join of all Pi gives
us a hyperspace P disjoint from v and they form a totally geodesic end for E.
(iii) From this, we obtain that g ∈ Zl is clearly diagonalizable with positive eigenvalues associated with Pi and x and the eigenvalue at x is different from ones at Pi .
We find gi0 ∈ Zl so that gi0 has the largest eigenvalue λ0i with respect to Ci as an
automorphism of RPvn−1 . Let λ0i,0 denote its eigenvalue at v as a projective automorphism of RP n . Then λ0i > λ0i,0 since it acts on the lens part not containing v. If Γi
is a trivial group, then we let gi be the identity. Each Ci0 ∩ Pi has an attracting fixed
point of gi ∈ Γi restricted to Pi if Γi is hyperbolic. We can choose gi so that the largest
eigenvalue λi of gi is sufficiently large. This follows since Γi is linear on Si − Pi where
we know that this is true for strictly convex cones by the theories of Koszul and so on.
Then by taking λi sufficiently large, gi0 gi has an attracting fixed point in Ci0 ∩ Pi . This
must be in Cl(Õ). By irreducibility, we obtain that all Ci0 ∩ Pi ⊂ Cl(Õ).
Let Di0 denote Ci0 ∩ Pi . Then the join D of Cl(D10 ), .., Cl(Dl0 ) equals P ∩ Cl(Õ), which
is h(π1 (E))-invariant. Then Do is a properly convex subset. If any point of Do is in
bdÕ, then D is a subset of bdÕ, and it is not possible to find a lens for the end.
Therefore, Do ⊂ Õ.
If the eigenvalue at x of g ∈ Zl is largest or smallest ones of multiplicity one, we
can find an open segment s meeting P in Õ. Acting by g i (s) for i ∈ Z, we obtain a
segment of length π in Cl(Õ). This contradicts the proper convexity. If the eigenvalue
at x of g ∈ Zl is the largest or smallest of multiplicity at least two, then we have Si
where g acts as the identity, which was ruled out above.
(iv) The hyperspace P separates Õ into two parts, ones in the end neighbohood U
and the subspace outside it. Clearly U covers E times an interval by the action of
h(π1 (E)) and the boundary of U goes to a compact orbifold projectively diffeomorphic
to E.
Denote by Di0 the set Ci0 ∩ Pi for each i = 1, .., l. Let Λi be ∂Di0 where Γi acts
irreducibly. Let Λ be the join of Λ1 ,..., Λl in P . From this, it follows that Zl acts on
the corresponding set Λ0 in the boundary of Ẽ in RPvn−1 is the end points of recurrent
geodesics in Ẽ: There is a projection Π sending RPvn−1 to RP l given by sending each
Ci to a point ei . This sends the join of Di ∩ P to the join of e1 , ..., el , an l − 1-simplex
∆. and Zl acts naturally and cocompactly on ∆ with induced maps so that Zl acts so
that the set of boundary points of recurrent geodesics in ∆o /Zl is dense in ∂∆.
Let z be a point outside Cl(U ) in Õ close to E. Take a convex hull of the orbit
h(π1 (E))(z) of z and suppose that there exists a maximal segment l in the boundary
surface S 0 of the convex hull which does not end at interior points of S 0 . Since S 0 meets
each ray from v at a unique point, it follows that l ends at x0 in the union of rays from
x passing Λ from v outside P or on P . If x0 is not in Λ, then by convexity, we have
a segment s from v ending at x0 and so meets P . By using Zl -action and (iii), we see
that for some sequence of gi ∈ Zl , gi (so ) converges to a segment of length π, which is
a contradiction. Thus l ends at a point of Λ.
33
Suppose that l ends at a point of Λ. Then l project to a compact surface S 0 /h(π1 (E))
an infinite ray which accumulates to a line with both end points in Λ by above. Thus,
an interior point of P is in S 0 . By convexity, S 0 has to be in P . This contradicts our
choice of z.
We see that there is no straight segment in the boundary S 0 of the convex hull. we
obtain a convex hypersurface by smoothing with boundary Λ. We can take a point z 0
in U nearer to v to obtain a convex hypersurface in U with boundary Λ.
As in the proof of Theorem 4.15, we see that this implies that E is strictly lensshaped.
(v) This follows by considering each irreducible parts of the lens.
Theorem 4.17. Let Hom(π1 (E), PGL(n + 1, R)) be the space of representations of the
fundamental group of n-orbifold E. Then the subspace of lens-shaped representations
is open. Hence, if π1 (E) satisfies the assumptions of Theorems 4.15 or 4.16, then the
subspace of strict lens-shaped representations is open.
Proof. Let µ be a representation π1 (E) → PGL(n + 1, R) acting on a strictly convex
(n + 1)-domain K bounded by two open n-cells A and B and bdK − A − B is a
nowhere dense set. We assume that A and B are smooth and convex. We note
that K/µ(π1 (E)) is a compact manifold with boundary with two closed n-orbifold
components A/µ(π1 (E)) ∪ B/µ(π1 (E)). We see that A and B are strictly convex
hypersurfaces. By using the theory of deformations of geometric structures on compact
orbifolds we obtain a manifold N 0 diffeomorphic to K/µ(π1 (E)). Ñ 0 is a manifold with
two boundary components A0 and B 0 and developing into RP n−1 . Suppose that µ0 is
sufficiently near µ. Then µ0 must act on A0 and B 0 sufficiently near in compact open
C 1 -topology.
Sufficiently small changes in the holonomy is reflected in a sufficiently small changes
in the projective connections and vice versa.
Since K is properly convex, a linear cone K 0 over K has a smooth strictly convex
hessian function V by Vey’s work [52]. In the fundamental domain of K 0 under the
action of µ(π1 (E) extended by a transformation γ : v 7→ 2v. The hessian has a lower
bound. If we change K to K 0 by sufficiently small changes in the affine connection,
the positive definiteness of the hessian in the fundamental domain and the boundary
fundamental domain is unchanged. Thus K 0 is also properly convex domain by Vey’s
work [52].
Thus K 0 is a properly convex domain with strictly convex boundary A0 and B 0 . The
complement Λ = Cl(K 0 ) − A0 − B 0 is a closed subset. Then by Theorem 4.15 and 4.16,
the end is also strictly lens-shaped.
Remark 4.18. Two end neighborhoods are simply-equivalent if one is a subset of the
other. We will use the equivalence relation generated by this.
If a hyperbolic group is acting on a lens-shaped cone in an affine space, then the cone
is uniquely determined up to the plus or minus identity transformations. This follows
since the attracting and repelling fixed-points are dense in the boundary of the lens
corresponding to the cone and the holonomy irreducible by the result of Yves Benoist
34
[7]. (If there are more than two fixed points, then the space of subspaces containing the
two fixed points gives us reducibility) We see that if a holonomy realizes a lens-shaped
admissible end, then by unique decomposition part of Theorems 4.15 and 4.16, every
other realized lens-shaped admissible end is actually projectively isomorphic; that is,
Ci → ±Ci (Usually realized as minus identity for some part of the decomposition and
identity for other parts of the decomposition.)
The uniqueness is clear for horospherical ends up to equivalences in RP n .
4.3.3. Examples. From hyperbolic manifolds, we can obtain some examples of ends.
Let M be a complete hyperbolic manifolds with cusps. M is a quotient space of the
interior Ω of a conic in RP n . Then the horoballs form the horospherical ends.
Suppose that M has totally geodesic imbedded surfaces S1 , .., Sn homotopic to the
ends. Then π1 (Si ) fixes a point outside the conic. Then Si acts on a lens-shaped
domain that is -neighborhoods of Si in Ω/π1 (M ). Hence, we can add the cone over
the lens-shaped domain to M to obtain the examples of real projective manifolds with
radial ends.
Proposition 4.19. Suppose that M is a topologically tame properly convex real projective orbifold with radial ends with admissible end fundamental groups. Assume that
M is not covered by a real line times a compact (n − 1)-orbifold. Suppose that each
end fundamental group is generated by closed curves about singularities or has the holonomy fixing the end vertex with eigenvalues 1. If each end is either horospherical or
has a compact totally geodesic properly convex hyperspace in end neighborhoods, then
the ends are admissible.
Proof. Let E be an end with a compact totally geodesic hyperspace Σ. Then there is
an end neighborhood U in the universal cover M̃ of M containing the universal cover
Σ̃ of Σ.
If each end fundamental group is generated by closed curves about singularities, then
since the singularites are of finite order, the eigenvalue corresponding the end vertex is
1. Now assume that the holonomy of the elements of the end fundamental group fixes
the end vertex with eigenvalues equal to 1.
Then U can be identified with a properly convex cone in an affine subspace Rn and
the end fundamental group acts on it as a discrete linear group of determinant 1. Then
the theory of convex cones apply and using the level sets of the Koszul-Vinberg function
we obtain a smooth convex one-sided neighborhood in U using Lemma 6.5 and 6.6 of
Goldman [32]. Also, the outer one-sided neighborhood can be obtained by a reflection
about the plane containing Σ̃ and the end vertex.
We raise a question that given a properly convex end, if for each holonomy, the
eigenvalues of the cone point is strictly between the smallest and largest absolute
values of the eigenvalues of the holonomy, then the end is lens-shaped.
Proposition 4.20. Let O be a 3-orbifold with the end orbifolds each of which is homeomorphic to a sphere S3,3,3 with three singularities of order 3. Then the orbifold has
admissible ends.
35
Proof. We can easily show that an end orbifold admits a complete affine structure or
is a quotient of a properly convex triangle as it cannot be a quotient of a half-space
with a distinguished foliation by lines. (The deformation space is a cell of dimension
two by the same method as in [23].) Take a finite-index free abelian group A of rank
two. Consider an end, i.e., a cone over such an orbifold with end vertex v.
In the complete affine structure case, A acts without fixed point on R2 in RP 2 and
on a properly convex domain Õ and on the unique supporting hyperspace H at the
end vertex v.
Suppose that A acts on R2 properly discontinuously. Then we can classify these to
show that the eigenvalues are 1 as a restricted transformations of H. (Here, we have
to normalize eigenvalues as the transformation on the projective space H and they
product to 1.) Since A acts with eigenvalue 1 at v, the eigenvalues of elements of A
are all 1. By Proposition 4.13, the end is horospherical.
In the triangle case, A acts with an element g 0 with an eigenvalue > 1 and an
eigenvalue < 1. This means that there are fixed points v1 and v2 other than v in a
direction of the vertices of the triangle in the cone. Then g 0 has four fixed points and
an invariant subspace P disjoint from v. It follows that the end fundamental group
acts on P as well. We have totally geodesic ends and by Theorem 4.16, the end is
lens-shaped.
4.4. The Hilbert metric on O. A Hilbert metric on an IPC-structure is defined as
a distance metric given by cross ratios. (We do not assume strictness here.) Let Ω be a
properly convex domain. Then dΩ (p, q) = log(o, s, q, p) where o and s are endpoints of
the maximal segment in Ω containing p, q. This gives us a well-defined Finsler metric.
Given an IPC-structure on O, there is a Hilbert metric dH on Õ and hence on Õ. This
induces a metric on O.
Let U 0 be the union of horospherical neighborhoods of ends of O and let Ũ 0 be the its
inverse image in Õ as in the above section. We choose mutually disjoint neighborhood
for each end and O − U 0 is a compact suborbifold K with boundary ∂U 0 . We will
choose sufficiently large K. Let K1 denote a fundamental domain of K.
We define Õ/Ũ 0 as the space obtained by collapsing each component of Ũ 0 to a point.
(This is might be different from the usual definition in topology but we use it here for
convenience.) That is each component of Ũ 0 is collapsed to a point to be denoted as an
end. More precisely, we remove the interior of Ũ 0 and produce the path metric using
the Hilbert metric and collapse each component of the boundary to a point. We still
call this the electric Hilbert metric and denote it by dU .
Finally, we can make each component of U smaller so that the minimum distances
between two components of Ũ is at least 1 under the Hilbert metric. —(*).
We note that given any two points x, y in Cl(Õ), there is a geodesic arc xy with
endpoints x, y so that its interior is in Õ. This is simple since we can use a straight line
and modify if it passes the components of Ũ but this does not change the properties.
Proposition 4.21. Let l be a straight line, i.e., a complete geodesic in the Hilbert
metric, γ andx be a point of Õ − l:
• There exists a shortest path m from x to γ that is a line.
36
• For any line m0 containing m and y ∈ m0 , the path in m0 from y to the point
of γ is one of the shortest segements.
Proof. The distance function f : l → R defined by f (y) = d(x, y) is a convex function
and f → +∞ as y tends to ends of l. Therefore, there is a connected compact interval
of minimum of f in l. Take a geodesic m from y to one of the points of the interval.
For any point on m, the shortest segment to l is again insider m since otherwise, we
get a shorter segment from x. For a point z on m0 but not on m, suppose that there is
a shortest geodesic m00 from z to l: Then m00 , m0 , l lies on a two-dimensional subspace
P and our discussion reduces to P ∩ Õ. The methods of Proposition 1.4 in [21] applies
and we are done.
The endpoint is called the foot of the perpendicular from x to γ. The foot is not
unique but we can show:
The set of foot from x in K̃ to a geodesic γ in K̃ is a connected subsegment in Õ.
4.5. Strict IPC-structures and the relative hyperbolicity of the metric spaces.
Definition 4.22. We will only study irreducible properly convex real projective structures on O, i.e., properly convex structures with irreducible holonomy representations
and convex ends.
We also need a condition that a straight arc in the boundary of Õ must be contained
in the closure of some end neighborhood of an end-vertex and as a consequence any
triangle with interior in Õ and boundary in bdÕ must be inside an end-neighborhood.
We call these two conditions no edge condition. The IPC-structure satisfying the no
edge condition is said to be the strict IPC-structures.
For some orbifolds, (n − 1)-dimensional convex real projective structures on end
orbifolds are always properly convex: If O is closed and π1 (O) is virtually trivial
center, then this holds. For example, closed 2-orbifolds of negative Euler characteristic
or closed orbifolds admitting hyperbolic or at least one IPC-structures by Benoist [6].
For orbifolds with ends, we can prove this in some cases at the moment without no
generality is in sight. For closed 2-orbifolds with zero Euler characteristic, this is
false. For example, consider tori and those covered by them such as a turn-over with
singularities of orders 3, 3, 3.
4.5.1. The convex hulls of ends. A convex hull of a subset A of RP n is defined as the
smallest closed subset containing A. By the compactness of RP n , a convex hull of a
set A is a union of the set S1 of 1-simplicies with endpoints in the closure of A and the
set S2 of 2-simplices with boundary edges in S1 and Si of i-simplices with boundary
sides in Si−1 for i = 3, 4, . . . , n. We denote it by CH(A).
One can associate a convex hull of an end E of Õ as follows: For horosphericalends,
the convex hull of each is defined to be empty actually.SFor a lens-shaphed end E with
an end vertex v, we define the convex hull to be CH( s∈S(x) s) ∩ Õ.
Proposition 4.23. Suppose that O is a topologically tame strictly IPC-orbifold. Let
E be a lens-shaped end and v an associated end vertex.
37
(i) A segment in the boundary of Õ is always contained in the closure of some
convex hull I and more precisely it is in the union S(v) of the segments in
bddev(Õ) ending at v for the corresponding v. Thus, the segment is contained
in the closure of any end neighborhood of v.
(ii) I is contained in CH(Cl(U1 )) ∩ Õ for any end neighborhood U1 of v.
(iii) Any segment in S(v) correspond to the boundary of Rv (E).
(iv) bdI ∩ Õ is contained in the union of a lens part of a lens-shaped end neighborhood.
(v) I contains the concave end-neighborhood of E and actually equals CH(Cl(U )) ∩
Õ for a concave end neighborhood U of v. Thus, I has nonempty interior.
(vi) Each segment from v maximal in Õ meets the set bdI ∩ Õ exactly once and
bdI ∩ Õ/Γv is a topological orbifold isotopic to E.
(vii) there exists a nonempty-interior convex hull I of a neighborhood of the end
vertex v of E of Õ and where Γ(v) acts so that I/Γ(v) is diffeomorphic to the
end orbifold times an interval.
(viii) I ∩ Õ has a boundary restricting to the covering map is an immersed compact
orbifold homotopic to the associated end orbifold.
Proof. (i) A segment in bdÕ is contained in a closure of an end neighborhood by the
strictness of the IPC-structure. Since it meets a segment in S(x), the segment must
be in the union of S(v) as in the proof of Theorems 4.15 and 4.16. (See (***)).
By Theorems 4.15 and 4.16, the set S(v) is always contained in the closure of any
end neighborhood of v. Thus (ii) follows.
(iii) A segment s from v in ClÕ either ends in a lens-shaped domain or is in bdÕ.
In the second case, s is in bdRv (E) clearly.
(iv) We define S1 as the set of 1-simplicies with endpoints in segments in S(v) and
we inductively
define Si to be the set of i-simplies with
S
S boundary in Si−1 . Then I is a
union σ∈S1 ∪S2 ∪···∪Sn σ. Notice that bdI is the union σ∈S1 ∪S2 ∪···∪Sn ,σ⊂bdI σ since each
point of bdI is contained in the interior of a simplex which lies in bdI by convexity of
I. If σ ∈ S1 with σ ⊂ bdI, then its end point must be in an endpoint of a segment in
S(v) or is a subsegment of an element of S(x). If the interior of σ is in a segment in
S(v), then σ is in the union of S(v) by convexity of ClRv (E). Hence, if σ o meets Õ,
then σ o is the lens-shaped domain. Now by induction, we can verify (iv).
(v) Since I contains the segments in S(x) and is convex, and so does a concave end
neighborhood U , we see that bdU −I is empty. Otherwise, let x be a point of bdU ∩bdI
where some neighborhood in bdU is not in I. Then a supporting hyperspace at x of
the convex set I, meets some segments in S(x) in the interior. This is a contradiction
since I contains the segments entirely. Thus, U ⊂ I.
(vi) bdI ∩ Õ is a subset of a lens part of an end by (iii). Each point of it meets
a maximal segment from v in the end but not in S(x) at exactly one point since a
maximal segment must leave the lens cone eventually. Thus bdI ∩ Õ is homeomorphic
to (n − 1)-cell and the result follows.
(vii) This follows from (v) since we can use rays from x meeting bdI ∩ Õ at unique
points and use them as leaves of a fibration.
38
Bd I
I
S(x)
Figure 3. The structure of a lens-shaped end.
(viii) This agin follows from (vi).
An elliptic element of g is an element of π1 (O fixing an interior point of Õ.
Proposition 4.24. Suppose that O is a topologically tame strictly IPC-orbifold. Each
nonidentity and nonelliptic element g of π1 (E) has either two fixed points in bdÕ not
in the closure of the end neighborhoods or has fixed points in the closure of a concave
end of a lens-shaped end or has unique fixed point at horospherical end vertex and is
in an end fundamental group. We conclude that π1 (E) for each end E is a finitely
generated parabolic subgroup.
Proof. Let g ∈ π1 (O). If g has a fixed point at a horopherical end vertex, then g must
act on the horoball since the horoball is either sent to a disjoint one or sent to the
identical one. Hence g is in the end fundamental group. Similarly, if g fixes a point
of the closure U of a concave end, g(U ) and U meet at a point. If they meet at an
interior point of U or g(U ), then U = g(U ) since they are end neighborhoods and g
is in the end fundamental group. If they meet at a boundary point but U 6= g(U ),
then they meet at a boundary point of a lens L and a boundary point of g(L). This
implies that for an inner boundary A of L, g(A) and A meet in their closure. By a
metric considerations, there are sequences xi ∈ A and yi ∈ g(A) so that the sequence
of distances between them goes to zero. However, since A and g(A) cover compact
imbedded submanifolds respectively that are disjoint in O as well, there should be a
positive lower bound. This is a contradiction. Therefore, g is in the end fundamental
group always.
Suppose that an element g of π1 (O) is not homotopic to any element of an end
fundamental subgroup. Then by above g does not fix any of the above types of points.
Suppose that g fixes a unique point x in the closure of bdÕ and x is not in the closure
of ends as above: Then it is easy to see that g has only eigenvalues of unit norms and
39
hence there is a g-invariant horoball tangent to bdÕ at x. Take a line l(t) converging to
x as t → 0. Then we see that d(l(t), g(l(t)) → 0 by the horoball geometry. Moreover,
the image of l(t) in O is not in some end neighborhood eventually and hence, it follows
that there is a sequence ti so that d(l(ti ), g(l(ti )) → 0 and l(ti ) and g(l(ti )) have images
in the complement C of ends in Õ. Since C covers a compact manifold, there is a
positive lower bound on d(l(ti ), g(l(ti )). This is a contradiction.
Therefore g fixes at least two points s and l in bdÕ. By strict convexity, there exists
a line segment l in the interior of Õ connecting the two points. The two fixed point
must have the positive real eigenvalues that are largest and smallest absolute values of
the eigenvalues of g since otherwise we can then form a great segment by acting on a
segment transversal to l. In fact, any invariant line in the interior of Õ connects the
highest and lowest positive real eigenvalues.
Suppose that there is a third fixed point k in bdÕ, then the line segment connecting
it to the s or t must be in bdÕ by above. This contradicts the strict IPC-property
and hence, we have that there are exactly two fixed points in bdÕ of the positive real
eigenvalues that are largest and smallest absolute values of the eigenvalues of g.
Recall that a lens-shaped end can have many factors but sometime, the factor can
be a unique hyperbolic group.
Proposition 4.25. Given a topologically tame IPC-orbifold O and its universal cover
Õ, there exists a collection of mutually disjoint open concave end-neighborhood for each
end that is a lens-shaped cone. Equivariantly remove some of these with hyperbolic end
fundamental groups from Õ and we obtain a convex domain as the universal cover of
the resulting orbifold O1 . If O is strictly IPC with respect to all of its ends, then O1 is
strictly IPC with respect to remaining ends.
Proof. If O1 is not convex, then there is a triangle T in Õ1 with three segments s0 , s1 , s2
so that T − so0 ⊂ Õ1 but so1 − Õ1 6= ∅. Then a point of x ∈ so1 − Õ1 6= ∅ is in one of the
removed concave-open neighborhoods or is in bdÕ itself. The possibility cannot occur
since the neighborhood is open and the second possibilty implies that O is not convex.
These are contradictions.
Suppose that O is strictly IPC. Any segment in the boundary of the developing image
of O is a subset of the closure of end neighborhood of an end vertex. A hyperbolic
concave end-neighbohood has strictly convex boundary in the space of lines through
the end vertices and hence only straight segments are on a segment from the end vertex.
Thus, concave end neighborhoods of end vertices contain all such segments.
Since we removed concave end neighborhoods of the hyperbolic ends, any straight
segments in bdÕ lie in the closures of end neighborhood of an end vertex. Thus, O1 is
strictly IPC.
4.5.2. Bowditch’s method. We will use Bowditch’s result to show
Theorem 4.26. Let O be a topologically tame strictly IPC-orbifold with radial ends
and has no essential annuli or tori. π1 (O) is relatively hyperbolic group with respect
40
to admissible end groups π1 (E1 ), ..., π1 (Ek ) where 1, ..., k are so that each Ei are horospherical or lens-shaped Let Ũi be the inverse image of a neighborhood Ui of the end Ei
for i = 1, . . . , k. Then
• π1 (O) is relatively hyperbolic with respect to the end groups π1 (E1 ), ..., π1 (Ek ).
• If π1 (El ), .., π1 (Ek ) are hyperbolic for some 0 ≤ l < k (possibily some of the hyperbolic ones), then π1 (O) is relatively hyperbolic with respect to π1 (E1 ), . . . , π1 (El−1 ).
Proof. We show that π1 (O) is relatively hyperbolic with respect to the end groups
π1 (E1 ), ..., π1 (Ek ). Since the group is roughly isometric to Õ, we have the second
statement.
We use bdÕ with an open concave neighborhood of end vertices removed which
is the boundary of a convex domain. Thus, we obtain bdÕ1 . We now identify each
component of the boundary of the concave end-neighborhood into a point and also
the concave end-neighborhoods of pseudo-reduced ends and obtain the corresponding
collapsed metric. We denote this quotient space bdÕ1 / ∼ by B. We show that π1 (O)
acts on the set B as a geometrically finite convergence group.
We will use Theorem 0.1. of Yaman [54]:
We first show that the group acts properly discontinuously on the set of triples in
B. Suppose not. Then there exists a sequence of nondegenerate triples {(pi , qi , ri )}
converging to a distinct triple {(p, q, r)} so that pi = γi (p0 ), qi = γi (q0 ), and ri = γi (r0 )
where γi is a sequence of mutually distinct elements of π1 (O). Here p, q, and r are
representatives of points of ∂dev(Õ1 )/ ∼. By multiplying by some element Ri near I
in PGL(n+1, R), we obtain that Ri ◦Ti for each i fixes p, q, r and restricts to a diagonal
matrice with entries λi , δi , µi on the plane with coordinates so that p = e1 , q = e2 , r =
e3 .
Then we can assume that λi δi µi = 1 by restricting to the plane and up to choosing
subsequences and renaming, we can assume that λi ≥ δi ≥ µi > 0. Thus λi → ∞
and µi → 0 since otherwise these two sequences are bounded and this contradicts the
discreteness of the holonomy.
By strictness of convexity, the interiors of the segments pi qi , qi ri , and ri pi are in the
interior of dev(Õ1 ).
We claim that one of the sequence λi /δi or the sequence δi /µi are bounded: Suppose
not. Then these sequences are unbouded and taking a convex open neighborhood of
qi and applying Ti , we see that there exists a segment of length π in the closure of the
developing image. This is a contradiction to the proper convexity of the developing
image.
Suppose now that λi /δi is bounded: Now the segmentspi qi converges to pq whose
interior is in dev(Õ1 ). Then we see that pq must be in the boundary as these points
must be the limit points of points of squence of Ti (s) for some compact subsegments
s ⊂ p0 q0 ) by the boundedness of the above ratio. If we assume that δi /µi is bounded,
then we obtain a contradiction similarly.
This proves the proper discontinuity of the action on the space of distinct triples.
An end group Γx for end vertex x is a parabolic subgroup fixing x since the elements
in Γx fixes only the contracted set in B and since there are no essential annuli. By
Proposition 4.24, we see that the groups of form Γx are the only parabolic subgroups.
41
Also, (B − {x})/Γx is easily seen to be homeomorphic to the end orbifold and
therefore, compact. Hence, Γx are the bounded parabolic subgroups.
To show that let p be a point that is not a horospherical endpoint or a singleton
corresponding an lens-shaped end or a pseudo-reduced ends of B.
We show that p is a conical limit point. This will complete our proof. We find a
sequence of holonomy transformations γi and distinct points a, b ∈ ∂X so that γi (p) →
a and γi (q) → b for all q ∈ ∂X − {p}. To do this, we draw a line l(t) from a point of
the fundamental domain to p where as t → ∞, l(t) → p in the compactification. Since
l(t) does not end in an end neighborhood, there is a sequence {ti } going to ∞ so that
l(ti ) is not in any of the end neighborhoods Ũ1 ∪ · · · ∪ Ũk . Let p0 be the other endpoint
of the complete extention of l(t) in Õ. We can assume without generality that p0 is not
in the closure of any end neighborhood.
Since (Õ − Ũ1 − · · · − Ũk )/Γ is compact, we have a compact fundamental domain F
of Õ − Ũ1 − · · · − Ũk with respect to Γ.
We find points zi ∈ F so that γi (l(ti )) = zi for a deck transformation γi . Then by
taking subsequences γi−1 (p) → a for a point a and γ −1 (p0 ) also converges to another
point b not equal to a. Take any point q ∈ X − {p} and find a geodesic m from q to l
with the property that every point on m a shortest geodesic from the point to l lies in
m by Proposition 4.21. Let q 0 be the intersection of m to l. γi−1 (q 0 ) then converges to b
by a Hilbert metric consideration. The sequence of segments γi−1 (qq 0 ) has the property
that every point on it is a shortest geodesic from the point to l lies in γi−1 (qq 0 ). (See
Figure 4.)
Because of this property, for any sequence of points xi ∈ γi−1 (qq 0 ), we have that
the shortest geodesic from xi to γi−1 (l) lie in γi−1 (qq 0 ). However, if the γi−1 (qq 0 ) is not
exiting, then we can choose xi ∈ γi−1 (qq 0 ) and xi → x ∈ Õo so that the corresponding
sequence of distances on γi−1 (qq 0 ) is converging to ∞. However, as xi → x ∈ Õo ,
the sequence d(xi , li ) is bounded; a contradiction. Therefore, γi−1 (qq 0 ) is an exiting
sequence to any compact subset of Õ.
By choose a subsequence, the sequence converges to a point or a segment in the
boundary ∂ Õ1 contained in an end singleton.
For the final item, we remove the concave ends of selected hyperbolic ends and obtain
tame orbifold O2 with radial ends. Then by Proposition 4.25, O2 is strictly IPC with
respect to the remaining ends. Hence, the proof above still applies.
4.5.3. Converse. We will prove the partial converse to the above Theorem 4.26:
Theorem 4.27. Let O be a topologically tame IPC-orbifold with admissible ends without essential annuli or tori. Suppose that π1 (O) is relatively hyperbolic group with
respect to the admissible end groups π1 (E1 ), ..., π1 (Ek ) where Ei are horospherical for
i = 1, ..., m and lens-shaped for i = m + 1, ..., Ek for 0 ≤ m ≤ k. Let ∼ denote
the equivalence class of Õ collapsing each component of ends to a point. Equivalently
suppose that the quotient space Õ/ ∼ is a δ-hyperbolic space for some δ > 0.
• Assume that O is IPC. Then O is strictly IPC.
42
p'
l
q'
m
q
p
Figure 4. The shortest geodesic m to a geodesic l.
• Assume now that π1 (O) is a relatively hyperbolic group with respect to the admissible end groups π1 (E1 ), ..., π1 (Ek ) with all the hyperbolic end fundamental
groups not included. Then again if O is IPC, then O is strictly IPC with respect
to the corresponding collection of admissible ends.
Proof. Let Ũ denote the inverse image of end neighborhoods U1 , ..., Uk . We define Ũ 0
to be the developing image of the union of inverse images of U1 , . . . , Uk in Ω1 .
Let Um+1 , .., Uk be the lens-shaped ends. If some Ui is lens-shaped, then we removed
a concave neighborhood from Ui and we can let Ui be the boundary of the concave
neighborhood. (Actually it is no longer a neighborhood of the end now.) Then Ũi is a
union of boundary components. We assume this from now on.
We will use the Hilbert metric dH of Õ restricted to Õ1 from now on. Also, Ui for
i = n + 1, . . . , k denotes Cl(Ui ) ∩ Õ1 from now on. The quotient is a compact convex
surface in O1 .
Note that the convex hull of lens-shaped neighborhoods are now contained in a lens
part of the neighborhoods.
Suppose that O is not strictly IPC.
We will obtain a triangle with boundary in ∂ Õ1 and not contained in the convex hull
of ends. Let l be a nontrivial maximal segment in ∂ Õ1 not contained in the closure
of an end neighborhood. First, lo does not meet the closure of a horospherical end
neighborhood by Proposition 4.13. By Theorems 4.15 and 4.16 we have that if lo meets
the closure of a lens-shaped end neighborhood, then lo is in the closure. Therefore, l
meets the closures of end neighborhoods possibly only at its endpoints.
Let P be a 2-dimensional subspace containing l and meeting Õ1 outside Ũ . We
define ¯˜UP0 as the union of the closure of each component of Ũ 0 ∩ P .
By above, lo is in the boundary of P ∩ Õ1 . Draw two segments s1 and s2 in P ∩ Õ1
from the end point of l meeting at a vertex p in the interior of Õ1 . We choose a
sequence of points xi on a line m in P converging to a point x in the interior of l. Then
43
dH (xi , s1 ∪ s2 ) → ∞ by consideration in the Hilbert metric by looking at all straight
segment from xi to a point of s1 or s2 .
We can choose xi so that it lies outside Ũ as only endpoints of l are possibly in
the closures of end neighborhoods. Recall that there is a fundamental domain F
of Õ1 − Ũ under the action of π1 (E). Since lo is not contained in any lens-shaped
end neighborhood, we further require that xi is not in a single convex lens-shaped
neighborhood.
Now, by taking xi to the fundamental domain F by gi , gi can be chosen to be a
sequence of mutually distinct elements of π1 (O) as xi is in a distinct images gi (F ).
We choose a subsequence so that gi (T ) geometrically converges to a convex set, which
could be a point or a segment or a nondegenerate triangle. Since gi (T )∩F 6= ∅, and the
sequence ∂gi (T ) exits any compact subsets of Õ1 always while dH (gi (xi ), ∂gi (T ) → ∞,
we see that a subsequence of gi (T ) converges to a nondegerate triangle, say T∞ .
We show that by choosing xi carefully, we can choose T∞ not inside a convex hull
of an end: If ∂T∞ is in the closure of one of the end neighborhoods, then the end
neighborhood is of lens type, and it is contained in the convex hull I of the end. We
are following p|m in O for the covering p : Õ → O. Suppose that the limit points of
p|m is in a single end. Then it follows that m is eventually inside one end and m ends
at a point of the end. This means lo meets the closure of an end. This was ruled out
above. Since p|m enters and leaves the convex hull or the end neighborhoods infinitely
often, there is at least one component line C of a limit set of the image of p|l so that
each component of p−1 (C) is not in a single component. We choose a subsequence xi
so that p(xi ) is converging to a point C as well as the direction of p|m near xi to the
direction of C. Now T∞ contains a component line of p−1 (C) not in a convex hull of
an end. Thus, the chosen T∞ is not in any convex hull of an end. By the following
Lemma 4.28, this cannot happen and O is strictly IPC.
For the final item, let O2 denote the O1 with the equivariant collection of the (n−1)balls corresponding to all the hyperbolic ends no longer considered ends. Then O2
covers a tame orbifold with radial ends. π1 (O2 ) is hyperbolic with respect to the
remaining ends π1 (Ei ).
We obtain that O2 is strictly IPC with respect to the nonhyperbolic ends by the
same reason as the first item.
Lemma 4.28. Suppose that O is a topologically tame properly convex n-orbifold with
radial ends that are properly convex or horospherical and π1 (O) is relatively hyperbolic
with respect to its ends. O has no essential tori or essential annulus. Then every
triangle T in Õ with ∂T ⊂ ∂ Õ is contained in the closure of a convex hull of its end.
Proof. We will use the notations of the proofs in above Theorem 4.27 but we do not
know the lens-shapedness of the nonhorospherical ends. We will not use O1 here or
the concave end neighborhood. Let T 0 be a triangle not in the closure of a convex hull
of an end.
Now we will consider various possibilites for the triangle: Denote by L a component
of the triangle T 0 − Ũ 0 .
Suppose that there is at least one g 6= I so that g(L) = L. Let v be a vertex of L.
Then Lhgi corresponds to an annulus mapping into O. Let l be a maximal geodesic in
44
Lo so that l and g(l) bounds a fundamental domain of the annulus. Then notice that
there is an arc α̃0 connecting a point of l to g(l) mapping to a closed curve α0 . By a
similarity based at v, we form a parameter of disjoint closed curves αt for t ∈ R.
Then the Hilbert lengths of αt are bounded above. Either
• αt never leaves an end neighborhood entirely for t greater than some constant
c and less than some c0 .
• αt enters an end neighborhood entirely and leaves it for t greater than some
constant c or less than some c0 .
• αt never enters an end neighborhood entirely for all t.
In the first case, L must be homotopic into an end neighborhood. This is ruled out
above. In the second case, g is freely homotopic to the end fundamental group. Assume
that αt is not in an end neighborhood entirely for t > c. If αt is not in an end neighborhood for t > c, then it cannot enter another end neighborhood entirely; otherwise,
we obtain a free homotopy between the fundamental groups of the two ends. In this
case, there is a subsequence ti so that αti converges to a closed curve not contained
in any end neighborhood. We see that αti and αtj are homotopic for i, j sufficiently
large. This implies that there exists an essential torus. This is a contradiction. The
third case also show that there exists an essential torus, a contradiction.
We suppose that there exists no g 6= I so that g(L) = L from now on.
Suppose that L meets infinitely many horospherical ends and the sizes of L inter0
sected
S with these are not bounded. Then we can show that L or a leaf L in its closure
of g g(L) meet a horoball and its vertex in its closure. However, in the first case, this
gives an arc in L or L0 with one horospherical endpoint equal to an interior point of an
edge or a vertex of L or L0 . Both cases are ruled out by Proposition 4.13.
Thus, the size of horospherical ends intersected with L is bounded. Therefore, by
choosing horospherical ends sufficiently small, we may assume that L does not meet
any horospherical ends.
We can easily assume that L is not a subset of a convex hull of an end since we can
choose such a triangle as above.
We will use the theory of tree-graded spaces and asymptotic cones [26] and [49].
We remove neighborhoods of sufficiently small horospherical ends from Õ and call the
result Õ0 . Then Õ0 is quasi-isometric with π1 (O).
Recall that π1 (U1 ) for an end neighborhood U1 is isomorphic to a finite extension of
l
Z × Γ1 × · · · × Γk for hyperbolic groups Γi .
Suppose first that l ≥ k for all of the ends. The end fundamental groups are never
hyperbolic and hence has no cut points asymptotically.
Let (X, dX ) be a metric space. Given an ultrafilter ω over the set I of natural
numbers, an observation point x = (xi )ω , and sequence of numbers δ = (δi )i∈I satisfying
limω δi = ∞, the ω-limit limω (X, dX /δi )e is called the asymptotic cone of X. (See [35],
[36] and Definitions 3.3 to 3.8 in [26].)
We will fix an ultrafilter ω in I and δ from now on. We set an observation point
to be a constant sequence. By Theorem 9.1 of Osin and Sapire [49] and Theorem 5.1
of Drutu and Sapir [26] and Theorem 4.26, π1 (Õ) is asymptotically tree-graded with
respsect to the end fundamental groups π1 (Ui ). Thus Õ0 is asymptotically tree graded
45
with respect to the end neighborhoods. Let Õ∞ denote the asymptotic cone of Õ0
with the ω-limit of the end neighborhoods as pieces. Here a piece is a closed subset
satisfying certain properties in [26]. Let P denote the set of pieces.
But the existence of L gives us a subspace L0 isometric with L in the asymptotic
cone. The geodesics here are precisely the straight lines.
Each point of L has an end component of uniformly bounded distance from it as the
diameter of O − U is bounded. Hence, each point of L0 is in an element of a piece. (See
Definion 3.9 of [26].) –(*)
Since each end component has a convex neighborhood in a bounded distance away,
we see that each piece is also convex. Thus, a piece intersected with L0 is a convex
subset, possibly with nonempty interior. Let the set of the intersections of pieces with
L0 be denoted by P 0 .
Suppose that L0 is contained in a piece. This implies that there is a sequence of
compact sets Ki of points in L with dO (x, L1,i ) ≤ µi for a lens L1,i of a lens-shaped
end Ui,1 and dO (yi , ∂Ki ) = λi → ∞ for yi ∈ Ki so that λi /δi → ∞ and µi /δi → 0 and
yi ∈ Ki asymptotically converges in L0 . This implies λi /µi → ∞ also.
Since the Hilbert metric is convex and so is a lens, Ki is convex. We obtain that since
λi /δi → ∞, Ki eventually contains y0 . Therefore, we may let yi = y0 for convenience.
Therefore, there is a sequence of segments y0 ∈ s̃i ⊂ Ki with ∂si ⊂ ∂Ki in L so that
for the minimum distance li of s̃i from y0 to the end points of s̃i , we have li /δi → ∞.
We claim that this cannot happen as li is less than or equal to µi times some constant:
Suppose that L1,i is bounded away from one of the end points of the extending line
m of s̃i in ∂L. Let si ∈ s̃i be a point of ∂si ⊂ ∂Ki . Let ti be the end point of the
extending line m further away from si from y0 and let s0i be the other end point of
m. We take a supporting hyperspace Hi at the point ti Then we take a point outside
Õ and a subspace Pi of Hi of codimension 2 in RP n disjoint from Õ, which exists by
the proper convexity of Pi ∩ Cl(Õ). Then we take subspaces containing Pi and points
s0i , ti , si , y0 and take the log of cross ratio to find the distance dO (y0 , si ) = λi (See Figure
5.)
We also take a geodesic ˜li from si to L1,i of length µi and denote by fi the end point
of the extension of si not in the end of L1,i . Then we take the log of the cross ratio of
fi , si and the two other points in the closure of L1,i to obtain the distance µi . We can
replace fi with f 0 the intersection of the line containing ˜li with Hi . The corresponding
logarithm µ0i is smaller than or equal to µi .
Seeing things from Pi , as the two points in Cl(Li ) is bounded away and two points
y0 , s0i , we can easily show by computations that λi ≤ Cµ0i ≤ Cµi for some positive
constant C. This is a contradiction.
Suppose now that L1,i is not bounded away from the both ends of m in ∂L. Since it
cannot be that L1,i converges to some set containing m as this means an interior of an
edge of L meets the closure of an end, which means that L is in the convex hull of the
end by convexity, a contradiction on the assumption on L. Now, we can suppose that
L1,i have points converging to an end point of ∂L or that L1,i contains an end point of
m. Let si ∈ s̃i be a point of ∂si ⊂ ∂Ki away from that end from y0 . Then a similar
argument will show a contradiction.
46
f'
fi
y
si
0
s'i
t
i
L 1,i
y
0
si
ti
L
1,i
Figure 5. The diagram for Theorem 4.27.
Therefore, we conclude that L0 cannot be contained in one piece.
It is not possible for exactly two components P 0 contain L0 : If not, they intersect
in a segment or a convex set with nonempty interior or a nontrivial segment. This
contradicts the definition of the pieces.
As a consequence, let p1 , p2 be two points of the interior of L0 not in a single piece.
Then there exists a point p3 in general position so that no two of p1 , p2 , p3 are contained
in a common piece.
By taking a geodesics between two of p1 , p2 , p3 , we obtain a simple triangle ∆0 This
contradicts the definition of tree-graded spaces.
We conclude that there exists no triangle such as L and we showed the strict convexity of O.
4.5.4. The relative hyperbolicity of the Hilbert metric of IPC-structures.
Theorem 4.29. Let O denote a topologically tame IPC-orbifold with admissible ends
without essential annuli or tori.
• An IPC-structure on O with admissible ends is strictly IPC if and only if π1 (O)
is relatively hyperbolic with respect to the subgroups π1 (Ei ) for ends of O.
47
• A small deformation of a strict IPC-orbifold with respect to admissible ends to
an IPC-structure with admissible end remains strictly IPC.
• An IPC-structure on O with admissible ends with all the hyperbolic ends not
regarded as ends is strictly IPC after removing the concave neighborhoods of the
hyperbolic ends if and only if π1 (O) is relatively hyperbolic with respect to the
subgroups π1 (Ei ) for ends of O with corresponding hyperbolic end fundamental
groups dropped.
• A small deformation of a strict IPC-orbifold with respect to admissible ends
with all the hyperbolic ends removed to an IPC-structure with admissible end
remains strictly IPC with respect to the same collection of ends.
Proof. The first item is proved by Theorem 4.26 and Theorem 4.27. For the second
item, we see that small deformation of structures do not change the groups themselves.
The third item is again proved by Theorem 4.26 and Theorem 4.27. The fourth item
is again proved by the fact that the groups do change under small deformations.
5. Openness of the convex structures
Given a real projective orbifold with radial end, each end has an orbifold structure
of dimension n − 1 and inherits a real projective structure.
Let U and sU be as in Section 3.3.2. We define Def iE,U ,sU ,ce (O) to be the subspace of
Def E,U ,sU (O) with real projective structures with admissible ends and irreducible holonomy homomorphisms and define Def iE,ce (O) to be the subspace of Def E (O) with real
projective structures with admissible ends and irreducible holonomy homomorphisms.
These subspaces for O has end fundamental group condition by Remark 4.18.
We define CDef E,sU ,ce (O) to be the subspace of consisting of IPC-structures in
Def iE,sU ,ce (O). Suppose that O satisfies the more generally the convex end fundamental group conditions. Then we define CDef E,ce (O) to be the subspace of Def iE,ce (O)
consisting of IPC-structures. Also, we have CDef E,ce (O) ⊂ Def iE,ce (O).
We define SDef iE,sU ,ce (O) to be the subspace of consisting of strict IPC-structures
in Def iE,sU ,ce (O). Suppose that O satisfies more generally the convex end fundamental group conditions. Then we define SDef E,ce (O) to be the subspace of Def iE,ce (O)
consisting of strict IPC-structures. Also, we have SDef RP n ,E,ce (O) ⊂ Def iRP n ,E,ce (O).
Theorem 5.1. Let O be a topologically tame real projective n-orbifold with admissible
ends. Suppose that O satisfies the end fundamental group condition or more generally
the convex end fundamental group conditions, and suppose that O has no essential
homotopy annulus. In Def iE,ce (O), the subspace CDef E (O, ce) of IPC-structures is
open, and so is SDef E (O, ce).
We caution the readers that for orbifolds like this the deformation space of convex
structures may only be a proper (possibly open) subset of space of the representations.
By Theorem 5.1 and Theorem A, we obtain
Corollary 5.2. Let O be a topologically tame real projective n-orbifold with admissible
ends. Suppose that O satisfies the end fundamental group condition or more generally
48
the convex end fundamental group conditions, and suppose that O has no essential
homotopy annulus.
hol : CDef E,ce (O) → repE,ce (π1 (O), PGL(n + 1, R))
is a local homeomorphism. Furthermore, if O has a strict IPC-structure with admissible
ends, then
hol : SDef E,ce (O) → repE,ce (π1 (O), PGL(n + 1, R)).
Here, in fact, one needs to prove for every possible continuous sections.
This theorem is proved by Koszul and expanded to real projective structures by
Goldman [29] for the closed real projective manifolds. See also Benoist [6].
5.1. Proof of the convexity theorem. A convex open cone V is a convex subset of
Rn+1 containing the origin O in the boundary. We note that any segment containing
O with a point of the cone can be extended to an open ray in the cone containing the
point. A properly convex open cone is a convex cone so that its closure do not contain a
pair of v, −v for a nonzero vector in Rn+1 . Equivalently, it does not contain a complete
affine line in its interior.
A dual convex cone V ∗ to a convex open cone is a subset of Rn+1∗ given by the
condition φ ∈ V ∗ if and only if φ(v) > 0 for all v ∈ Cl(V ).
Recall that V is a properly convex open cone if and only if so is V ∗ and (V ∗ )∗ = V
under the identification (Rn+1∗ )∗ = Rn+1 . Also, if V ⊂ W , then V ∗ ⊃ W ∗ .
For properly convex open subset Ω of RP n , its dual Ω∗ in RP n∗ is given by taking a
cone V in Rn+1 corresponding to Ω and taking the dual V ∗ and projecting it to RP n∗ .
The dual is a properly convex open domain if so was Ω.
Recall Koszul-Vinberg function
Z
e−φ(x) dφ
(4)
fV ∗ (x) =
V∗
where the integral is over the euclidean measure in Rn+1∗ . This function is strictly
convex if V is properly convex. f is homogeneous of degree −(n + 1) and the Hessian
∂i ∂j log(f ) gives us an invariant metric. Writing D as the affine connection, we will
write the Hessian Dd log(f ). We claim that the hessian is a positive definite and strictly
bounded below in a compact subset K of Cl(V ) − {O}. (See [32].)
Lemma 5.3. Suppose that fV ∗ (x) is defined on a neighborhood of x ∈ V . Let V1 be
another properly convex cone containing the same neighborhood. Let V ∗ has projectivization Ω and the dual V1∗ of V1 has projectivization Ω1 . For given any > 0, there
exists δ > 0 so that if the Hausdorff distance between Ω and Ω1 is δ-close, then fV ∗ (x)
and fV1∗ (x) is -close in a sufficiently small neighborhood of x in C i -topology for i ≥ 0.
Proof. Suppose that Ω ⊂ Nδ (Ω1 ) and Ω1 ⊂ Nδ (Ω). We also have that Ω−Nδ (∂Ω) ⊂ Ω1
and Ω1 − Nδ (∂Ω1 ) ⊂ Ω. We now choose the neighborhood of x so that so that they are
in the duals of the cones corresponding to Nδ (Ω), Nδ (Ω1 ), Ω − Nδ (∂Ω), Ω1 − Nδ (∂Ω1 ).
Since the integral is computable from an affine hyperspace meeting V ∗ and V1∗ in
bounded precompact convex sets and e−φ(x) is uniformly bounded, we see that the
integrals and their derivatives are estimable from each other, the result follows. (See
the proof of Theorem 6.4 of [32].)
49
The first step is to find an “expansion of neighborhood of the end”.
Lemma 5.4. Let O have a topologically tame IPC-structure µ with admissible ends.
Let U1 be an admissible neighborhood of an end with the vertex v in Õ that is foliated
by segments from v.
• There exists a sequence of convex open neighborhoods Ui of U1 in Õ so that
Ui − Uj /Γv for a fixed j and i > j is homeomorphic to a product of a closed
interval with the end orbifold.
• Given a compact subset of Õ, there exists an integer i0 such that Ui for i > i0
contains it.
• We can choose Ui so that ∂Ui is smoothly imbedded and strictly convex.
• The Hausdorff distance between Ui and Õ can be made as small as possible.
Proof. The end neighborhood U1 is foliated by segments from v. The foliation leaves
are geodesics concurrently ending at a vertex v corresponding to the end of U1 . We
follow the foliation outward from U1 and take a union of finitely many geodesic leaves
L of finite length from U1 and take the convex hull of U1 and ΓE 0 (L) in Õ.
If U1 is horospherical, then the convex hull is again horospherical. We can smooth
the boundary to be convex. Call the set Ωt where t is a parameter → ∞ measuring
the distance from U1 . By taking L sufficiently densely, we can choose a sequence Ωi of
strictly convex horospherical open sets at v so that evenutally any compact subset of
Õ is in it for sufficiently large i.
If U1 is lens-shaped, then take its convex hull I of the end. Take a union of finitely
many geodesic leaves L of length t from I and take the convex hull of I and ΓE 0 (L) in
Õ. Denote the result by Ωt .
We claim that bdΩt ∩ Õ is a connected (n − 1)-cell and bdΩt ∩ Õ/Γv is a compact
(n − 1)-orbifold homeomorphic to E and a boundary compont of I bounds a compact
orbifold a boundary compont of I bounds a compact orbifold homeomorphic to the
product of a closed interval with bdΩt ∩ Õ/Γv : First, each leaf of g(l), g ∈ ΓE 0 for l
in L is so that any converging subsequence of g(l), g ∈ ΓE 0 converges to a segment in
S(v) as g takes infinitely many values. This follows since a limit is a segment in bdÕ
with an endpoint v and must belong to S(x) by Proposition 4.13.
Let S1 be the set of segments with end points in ΓE 0 (L) and define inductively Si be
the set of simplicies with sides in Si−1 . Then the convex hull of ΓE 0 (L) is a union of
S1 ∪ · · · ∪ Sn . For each maximal segment s from v not in S(x), it meets the boundary
of the convex hull at at most one point since we can take a convex hull of I and s.
Suppose that a maximal segment s not in S(x) does not meet the boundary of the
0
convex hull. Let v 0 be its other end point in bdÕ. Then vS
is contained in the interior
of a simplex σ in Si for some i. Then the vertex of σ is in t∈S(x) t as we can verify by
induction. However, this implies that σ is in the convex hull C and no interior point
is in bdÕ.
Therefore, each maximal segment s from v meets the boundary bdΩt ∩ Õ exactly
once. Hence, the claim follows similar to the proofs in Theorem 4.15.
50
By taking sufficiently many leaves for L with lengths t sufficiently large, we see that
any compact subset is inside Ωt for some t sufficiently large. From this, the final item
follows also.
Let O have an IPC-structure µ with admissible ends. Clearly Õ is a properly convex
open domain. Then an affine suspension of O has an affine Hessian metric Ddφ for a
function φ defined on the cone in Rn+1 corresponding to Õ by above.
Proposition 5.5. Suppose that O satisfies the end fundamental group condition or
more generally the convex end fundamental group conditions, and suppose that O has
no essential homotopy annuli or tori. Suppose that the O has two IPC real projective
structures µ1 and µ2 with admissible ends and that suspension of O with µ1 has a Hessian metric. Suppose that µ1 is strictly IPC, and the second structure µ2 is sufficiently
close to the first one in C 2 sense, then a suspension of µ2 also has a Hessian metric
and O.
Proof. Suppose we modify the projective structure µ1 to µ2 on O in C 2 -topology. Let
Õ denote the universal cover corresponding to µ1 and let Õ0 denote one corresponding
to µ2 . Again these are identified with subsets of RP n .
Since µ1 and µ2 are close in the C 2 -topology of the developing maps, given a compact
subset K of Õ, there is a corresponding K 0 in Õ0 that is -close in the Hausdorff
topology.
We will now work on one end at a time: Let us fix an end E of Õ. We may assume
that the vertex p of E and the perturbed vertex p of E is the same by post-composing
the developing map by a transformation near the identity. Let K be a compact subset
of Õ and K 0 the perturbed one in Õ0 . Also, a compact subset Rp (K) ⊂ R(E) is
changed to a compact Rp (K 0 ) ⊂ R(E 0 ).
We use a projective transformation of RP n very close to the identity map, to make
sure that Rp (K 0 ) ⊂ Rp (E) for the set of rays R(E 0 ) of the corresponding end E 0 of the
perturbed O.
Let r0 (A) denote the union of rays in a subset A of R(E). Then a smooth and strictly
convex Ωt as obtained by Lemma 5.4 is changed to a smooth and strictly convex Ω0t
close to Ωt in r0 (K 0 ) as we changed the real projective structures sufficiently small in
C 2 -sense and hence the holonomy of the generators of π1 (E) is changed by a small
amount.
A very thin space is a space which is an -neighborhood of its boundary. Let rv (K)
denote the union of great segments with vertex v passing through points of K. By
Corollary 4.6, we may assume that R(E) and R(E 0 ) are sufficiently close convex domains in the Hausdorff sense. Thus, (r(E) − r(K)) is a very thin space and so is
rv (E 0 ) − rv (K 0 ). Given an > 0, we can choose K and K 0 and small deformation of
the real projective structures so that Cl(Ωt ) ∩ (r(E) − r(K)) is in an -neighborhood of
Cl(Ωt ) ∩ r(K) and Cl(Ω0t ) ∩ (rv (E 0 ) − rv (K 0 )) is in an -neighborhood of Cl(Ω0t ) ∩ rv (K 0 ).
Therefore the Hausdorff distance between Cl(Ωt ) and Cl(Ω0t ) can be made as small as
desired.
51
By Lemma 5.4, the Hausdorff distance between Cl(Ω0t ) and Õ can be made as small
∗
as desired. By Lemma 5.6, their duals Cl(Ω0∗
t ) and Õ are also as small as desired.
0
Hence, the Hessian functions f defined by equation 4 on the perturbed and suspended
Cl(Ω0t ) is very close to the original Hessian function f in compact subsets of Ω0t in the
C 3 topology by Lemma 5.3.
Note that the deck transformation π1 (O) being in SL(n + 1, R), they preserve f and
f 0 under deck transformations.
Now do this for all ends and we obtain functions f 0 on the inverse image U of the
union of the neighborhoods of ends. Since there is a C 2 -map arbitrarily close to the
identity map on the set V equal to Õ with end neighborhoods removed to the perturbed
Õ with end neighborhoods removed, we transfer f to Õ(R) by this map. Denote the
result by f 00 .
We obtain f 0 sufficiently close to f on the compact part of U ∩V and f 0 (tv) = t−n f 0 (v)
and f (tv) = t−n f (v). We find a C ∞ map φ defined on U ∩V so that φ(tv) = t−n φ(v) and
f 0 (v) = φf 000 (v) and φ is very close to the constant value 1 function. Using a partition
of unity adopted to U and V , we form φ0 which differs from 1 only in a compact
neighborhood of ∂V ∩ U in U . By making f 0 sufficiently close to f 000 as possible in
derivatives up to two, we see that φ has derivatives as close to 0 in a compact subset
as we wish: For example, define φ0 = f 000 /f 0 in a compact neighborhood K of ∂V ∩ U
in U and be 1 outside it. This is accomplished by taking a partition of unity functions
p1 , p2 so that p1 = 1 on (U − V ) ∪ K and zero outside the neighborhood of (U − V ) ∪ K
and p1 + p2 = 1 identically and define
φ0 = (f 000 /f 0 − (1 − ))p1 + p2 + (1 − ) if f 000 /f 0 > 1 − in V ∩ U . Thus, using φ0 we obtain a Hessian function f 0000 obtained from f 0 and φ0 f 000
on V ∩ U and extending them naturally. We can check the welded function from f 0
and φ0 f 000 has the desired Hessian properties.
Recall the standard elliptic metric d of RP n . We also have the elliptic metric d on
RP n∗ , denoted by the same letter. Define the injectivity radius of a properly convex
domain ∆ is given as the maximum of d(x, bd∆) for x ∈ ∆ and d(y, ∂∆∗ ) and y ∈ ∆∗
for the dual ∆∗ of ∆.
Lemma 5.6. Let ∆ be a properly convex open domain in RP n and ∆∗ its dual in
RP n∗ . Let be a positive number less than the injectivity radius. Then the following
hold:
• N (∆) ⊂ (∆∗ − Cl(N (bd∆∗ ))∗ .
• If two properly convex open domains ∆1 and ∆2 is of Hausdorff distance < ,
then ∆∗1 and ∆∗2 of Hausdorff distance < .
• More precisely, if ∆2 ⊂ N0 (∆1 ) and ∆1 ⊂ N0 (∆2 ) for 0 < 0 < , then we have
∆∗2 ⊃ ∆∗1 − Cl(N0 (bd∆∗1 ) and ∆∗1 ⊃ ∆∗2 − Cl(N0 (bd∆∗2 ).
Proof. Using the double covering map Sn → RP n and Sn∗ → RP n∗ of unit spheres in
Rn+1 and Rn+1∗ , we take components of ∆ and ∆∗ . It is easy to show that the result
for properly convex open domains in Sn and Sn∗ is sufficient.
52
z'
y
z
Figure 6. The diagram for Lemma 5.6.
For the first item, let y ∈ N (∆). Suppose that φ(y) < 0 for φ ∈ Cl((∆∗ −
Cl(N (bd∆∗ ))) 6= ∅. Since φ ∈ ∆∗ , the set of positive valued points of Sn under φ
is an open hemisphere H containing ∆ but not containing y. The boundary of the
hemisphere has a closest point z in bd∆ of distance ≤ . The closest point z 0 on the
boundary is in N (∆) since y is in N (∆) − H and z 0 is closest to bd∆. The great
circle containing z and z 0 are perpendicular to the boundary of the hemisphere and
hence passes the center of the hemisphere. One can push the center of the hemisphere
on a great circle containing z and z 0 until it becomes a supporting hemisphere. The
corresponding φ0 is in bd∆∗ and the distance between φ and φ0 is less than . This is
a contradiction. Thus, ⊂ relation holds. (See Figure 6.)
For the final item, we have that ∆2 ⊂ N0 (∆1 ) and ∆1 ⊂ N0 (∆2 ) for 0 < 0 < .
Hence, ∆2 ⊂ (∆∗1 − Cl(N0 (bd∆∗1 ))∗ : Thus, ∆∗2 ⊃ ∆∗1 − Cl(N0 (bd∆∗1 ) and so N (∆∗2 ) ⊃
∆∗1 and conversely.
Suppose that O has a IPC-structure µ. We will show that sufficiently close structure
µ that has admissible ends is also IPC.
Let Os be the affine suspension of O with the universal cover Õs . Recall the projective completion Ô of O. This is a completion of Õ using the path metric induced from
the pull-back of the standard Riemannian metric on RP n . (See [16] for details.) Let
Ôs denote the completion of Õ under the standard Euclidean metric of Rn+1 .
Suppose that µ0 is not convex. Then there exists a triangle imbedded in Ôs with
points in the interior of edge in the limit set Ôs − Õs . We can move the triangle so
that the interior of an edge l has a unique point x∞ in Ôs − Õs and x∞ do not map to
the origin under the developing map. We form a parameter of geodesics lt , t ∈ [0, ] in
the triangle so that l0 = l and lt is close to l in the triangle.
Now, we can lift such a triangle and the segment to Ôs and they correspond under
the radial projection. We use the same notation for the lift and the image.
From above, we see that an affine suspension µ00 of µ0 also has a Hessian metric d0 .
We prove that µ00 is properly convex, which will show µ0 is properly convex:
There is a minimum length under d0 in any arc with end points in the suspension
0
U of the a neighborhood of the union of ends in Os but is not homotopic into the
0
53
neighborhood. Let us call this number k. Let Ũ denote the inverse image of the union
U of ends in Õs . Suppose that l0 − Ũ has infinitely many components of arcs that are
not homotopic into Ũ with their endpoints fixed. Then l0 − Ũ is infinitely long and the
length of lt goes to ∞ as t → 0 as lt approximates l in for any given sufficiently large
closed subarc of l0 as closely as possible. Moreover, we can assume that the direction
of lt are constant when developed.
Let p, q be the endpoints of l. Then the Hessian metric is D0 dφ for a function φ
defined on Õs . And dφ|p and dφ|q are bounded, where D0 is the affine connection of
µ00 . This should be true for pt and qt for sufficiently small t uniformly. Let s be the
affine parameter of lt , i.e., lt (s) is a constant speed line in Rn+1 when developed. We
assume that s ∈ (t , 1 − t ) parameterize lt for sufficiently small t where t → 0 as
t → 0 and dlt /ds = v for a parallel vector v. The function Dv0 dv φ(lt (s)) is uniformly
bounded since its integral dv phi(lt (s)) is strictly increasing by the strict convexity and
convergesR to certain values as s → t , 1 − t .
p
1−
Since t t Dv0 dv φ(lt (s))ds = dφ(pt )(v) − dφ(qt )(v), the function Dv0 dv φ(lt (s)) is
also integrable and have a bounded integral by Jensen’s inequality. This means that
the length of lt is bounded. If l met infinitely many components of Ũ , then the length
is infinite since the minimum distance between components of U is bounded below.
As t → 0, this is bounded, we conclude that l can be divided into finite subsections
each of which meets one component of U and any subarc of it with end points in U is
homotopic into that component with endpoints fixed.
Now we go back to Õ. As above denote by Ũ the inverse image of the union of end
neighborhoods in Õ. Let s be the subsegment of l containing x∞ and meeting only
one component U1 of Ũ with bds ∈ bdU1 . Let st be the subsegment of lt so that the
endpoints of st converges to those of s as t → 0. Let p0 and q 0 be the endpoint of s.
Suppose that x∞ is the end of the radial lines of U1 and s should concide with radial
lines. Since p0 is in U1 and the set of rays in U1 form a convex subset of the projective
space of rays, it follows that q 0 is outside U1 . This is clearly a contradiction.
Now suppose that x∞ is in the middle of the radial line. Then the interior of the
triangle is transversal to the radial lines. Since our end orbifold is convex, there cannot
be such a line with a single interior point in the ideal set. This is again a contradiction
and Õ is convex.
Finally, for sufficiently small deformation, the convex real projective structures are
properly convex. If not, then there are sufficiently small deformed convex real projective
structures which are not proper and hence their holonomy is reducible. By taking
limits, the original one has to be reducible as well. However, we assume that it was
irreducible. Since the subspace of reducible representation is closed, we see that there
is an open set of irreducible properly convex projective structures near the initial one
µ.
Suppose now that O is strictly IPC. The relative hyperbolicity of Õ with respect to
the ends is stable under small deformations since it is a metric property of the large
scale invariant under quasi-isometries by Theorem 4.29.
This completes the proof of Theorem 5.1 and hence Corollary 5.2.
54
6. The closedness of convex real projective structures
Let us denote by repiE (π1 (O), PGL(n + 1, R)) the subspace of irreducible representations. which is well-known to be an open dense subset of repE (π1 (O), PGL(n + 1, R))
repiE,ce (π1 (O), PGL(n + 1, R)) the subspace of irreducible representations. which is
well-known to be an open dense subset of repE,ce (π1 (O), PGL(n + 1, R))
An end is permanently properly convex if the fundamental group is of form Zl × Γ1 ×
· · · × Γk where k = l ≥ 1 or if the group is hyperbolic. In this case, the end orbifold
is always properly convex by the theory of Benoist since each hyperbolic factor groups
have this property. (See Theorem 4.15.))
Theorem 6.1. Let O be a topologically tame IPC n-dimensional orbifold with convex end fundamental group condition. Assume that π1 (O) has no nontrivial nilpotent
normal subgroup. Then the following hold:
• The deformation space CDef E,ce (O) of IPC-structures on O maps homeomorphic to the union of components of repiE,ce (π1 (O), PGL(n + 1, R)).
• Furthermore, if the ends of O are permanently properly convex, then hol maps
the deformation space of IPC-structures on O homeomorphic to a union of
components of repiE,ce (π1 (O), PGL(n + 1, R)).
• Suppose also that O has no essential homotopy annulus or torus. Similarly, the
same can be said for SDef E,ce (O) of strict IPC-structures on O.
Proof. We first show that hol is injective on CDef E,ce (O). If not, then there exists a
homomorphism h : π1 (O) → PGL(n + 1, R) and properly convex convex domains Ω1
and Ω2 where h(π1 (O)) acts properly on so that Ω1 /h(π1 (O)) and Ω2 /h(π1 (O)) are
both diffeomorphic to O by a diffeomorphism inducing h.
Suppose that Ω1 and Ω2 are distinct . We claim that Ω1 and Ω2 are disjoint. Suppose
not. Then let Ω0 be the intersection where Γ := h(π1 (O)) acts. Each end fundamental
group also acts on Ω0 also. We can form a topological space Ω0 /Γ with end neighborhood
system. Since Ω1 , Ω2 , Ω0 are all n-cells, we see that Ω1 /Γ, Ω2 /Γ, and Ω0 /Γ are all
homotopy equivalent relative to the end. By taking a torsion-free index subgroup, we
see that then Ω0 /Γ is a closed submanifold in Ω1 /Γ and in Ω2 /Γ. The map has to be
onto in order for the map to be a homotopy equivalence. We see that Ω0 = Ω1 = Ω2 .
Suppose now that Ω1 and Ω2 are disjoint. Each corresponding pair of of the end
neighborhoods share end vertex since Ω1 and Ω2 are disjoint but each pair of the ends
have same end holonomy groups. Now Cl(Ω1 ) ∩ Cl(Ω2 ) is a compact properly convex
subset K of dimension < n and is not empty since the fixed points of the ends are in
it. The minimal hyperspace containing K is a proper subspace and is invariant under
h(π( O)). This contradicts the irreducibility.
Hence, this proves that hol is injective. The fact that hol is an open map was
discusses in Theorem 5.1.
Suppose that a convex domain is not proper. Then by [14], such a domain contains
a great sphere of dimension ≥ 0 in its boundary. If a representation h acts on this set,
then h is reducible.
To show that the image is of hol is closed. We show that the image of hol0 :
CDef 0RP n ,E (O) → HomiE (π1 (O), PGL(n + 1, R)) is closed. Let (Ωi , hi ) be a sequence
55
of a pair of convex domain and holonomies so that hi → h. The limit h is a discrete
representation by Lemma 1.1 of Goldman-Millson [34]. The closure of Ωi also converges
to a properly convex domain Ω since h(π1 (O)) acts on Ω and Ω being not properly
convex implies that h is reducible.
Since h is irreducible and acts on Ωo properly discontinuously, it follows that Ωo /h(π1 (O))
is an orbifold O1 homotopy equivalent to O. O1 can be purturbed to O10 so that it has
a holonomy h0 that is a holonomy of real projective structure µ on O. The domain
Ω1 of h0 has to equal Ω0 of the projective structure µ on O by the above uniqueness.
Hence, it follows that O1 is projectively diffeomorphic to O with µ and hence O1 is
diffeomorphic to O.
This show that the image of hol is closed.
The final statement is obvious.
Remark 6.2. We question whether the components of HomiE,ce (π1 (O), PGL(n + 1, R))
containing convex real projective structures with radial ends are really just a component
of HomiE,ce (π1 (O), PGL(n + 1, R)) under some conditions on O. For closed orbifolds,
this is proved by Benoist [2].
The developing image Ω as above is convex but may not be properly convex. Since
the limit holonomy h is bounded, we see that that our situation is as in [14]. Here,
Ω/h(π1 (O) admits a totally geodesic foliation by affine space with transverse projective structure. The space of leaves in the universal cover has a transverse projective
structure a properly convex domain. We do not yet understand these completely.
7. Examples
7.1. Main examples. For dimension 2, the surfaces with principal boundary component can be made into surfaces with ends since we can select the fixed points for each
boundary components and produce radial ends.
Let P a a properly convex polytope in an affine patch of RP n with faces Fi where
each side Fi ∩ Fj of codimension two is given an integer nij ≥ 2. A reflection orbifold
based on P is given by a group of reflections Ri associated with faces satisfying relations
(Ri Rj )nij = I for every pair i, j with Fi ∩ Fj is a side of codimension 2. Vinberg showed
that Ri acts on a convex domain in RP n with fundamental domain P and thus P with
a number of vertices removed can be given a structure of orbifold with interior of faces
Fi silvered and the interior of edge is given dihedral group structure and so on.
Then the theories of this paper is applicable to such orbifolds. Then the deformation
theorems obviously holds. The convexity also holds since they are reflection groups
according to Vinberg’s work (See [18].) When P is a dodecahedron or cube with all
edge orders 3, then we obtain an orbifold with radial ends A computation done by G.
Lee shows that there are nontrivial deformations where ends deforms.
The example of S. Tillman is an orbifold on a 3-sphere with singularity consisting of
two unknotted circles linking each other only once under a projection to 2-plane and a
segment connecting the circles (looking like a handcuff) with vertices removed and all
arcs given cyclic group of order three. This is one of the simplest hyperbolic orbifolds in
D. Heard, C. Hodgson, B. Martelli, and C. Petronio [37]. The end orbifolds are two 2spheres with three singularities of orders equal to 3 respectively. These orbifolds always
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have convex real projective structures in dimension 2, and real projective structures on
them has to be convex which is either a quotient of a properly convex open triangle or
a complete affine plane.
Since this orbifold admits a complete hyperbolic structure, (Õ, Ũ ) admit hyperbolic
metric for U the cusp neighborhoods. The complete hyperbolic structure is a strict
IPC-structure.
He found a one-dimensional solution set from the complete hyperbolic structure by
explicit computations. His main questions are the preservation of convexity. That
small deformation remain strictly IPC as given by our theory.
The another main example can be obtained by doubling a reflection orbifold based
on a convex polytopes. Here the P is doubled to an n-sphere with singularities are the
closure of the union of the sides whose interior has cyclic group of orders nij .
A double of tetrahedral orbifold of orders 3 is a double of the reflection orbifold
based on a convex tetrahedron with orders all equal to 3. This also admits a complete
hyperbolic structure since we can take the two tetrahedra to be the regular complete
hyperbolic tetrahedra and glue them by isometries. The end orbifolds are four 2-spheres
with three singular points of orders 3. We will study this objects. We computed that
the deformation space of real projective structures of a smooth local deformation space
of dimension 3 using exact algebra computations.
7.2. Nonexistence of reducible holonomy. One question is whether we can avoid
using irreducibilty; that is, given a strict IPC-structure, its deformation is always
strictly IPC without converging to reducible cases. This was shown to be true for closed
surfaces of negative Euler characteristic by Goldman [29], and for closed manifold with
hyperbolic fundamental group by Benoist [6] (for closed hyperbolic 3-manifolds by
Inkang Kim [45]).
We cannot do this for all cases but we accomplished it in a particular case. But we
think that these type of results should be true for wide class of orbifolds admitting
negatively curved Riemannian metrics.
The following theorem applies to S. Tillman’s example and the doubled tetrahedral
orbifold. These orbifolds having a complete hyperbolic structure has no essential annulus. We can try to generalize the theorem slightly but we do not attempt it since
there might be more complete generalization.
Maybe our cleanest result is the following. We will generalize the following theorem
in later papers.
Theorem 7.1. Suppose that a 3-dimensional orbifold is triangulated into one or two
tetrahedra with edges in the singular locus and the vertices are all removed. Suppose
that this orbifold has no essential homotopy annulus or equivalently it admits a complete
hyperbolic structure. The end orbifolds have Euler characteristic equal to zero and all
the singularities are of order 3. Then we have
(5)
Def E (O) = SDef E,ce (O).
and hol maps Def E (O) as an onto-map a component of representations
repE (π1 (O), PGL(4, R))
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which is also a component of repiE,ce (π1 (O), PGL(4, R)).
Proof. In [15], we showed that the boundary real projective structure determined the
real projective structure on O. First, there is a map Def E (O) → Def(bdO) given
by sending the real projective structures on O to the real projective structure of the
ends. (Here if bdO has many components, then Def(bdO) is the product space of the
deformation space of all components.) Let J be the image.
The end orbifold is so that if given an element of the deformation, then the geodesic
triangulation is uniquely obtained and there is a proper map from Def E (O) to the
space of invariants of the triangulations as in [15], i.e, the product space of cross-ratios
and Goldman-invariants spaces. (The projective structures are bounded if and only if
the projective invariants are bounded.) Thus by the result of [15], there is an inverse
to the above map s : J → Def E (O) that is a homeomorphism.
We can derive from the result of Goldman [29] and Choi-Goldman [23] that given
projective invariant ρ1 , ρ2 , ρ2 , σ1 , σ2 satisfying ρ1 ρ2 ρ3 = σ1 σ2 we can determine the
structure on S3,3,3 completely. We can assign invariants at each edge of the tetrahedron and the Goldman σ-invariants at the vertices. This is given by starting from
the first convex tetrahedron and gluing one by one using the projective invariants
(see [15]): Let the first one by always be the standard tetrahedron with vertices
[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], and [0, 0, 0, 1] and we let T2 a fixed adjacent tetrahedron with vertices [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0] and [2, 2, 2, −1]. Then by projective
invariants will determine all other tetrahedron triangulating Õ. Given any deck transformation γ, T1 and γ(T1 ) will be connected by a sequence of tetrahedrons related by
adjacency and their pasting maps are completely determined by the projective invariants – so-called holographic cases. Therefore, as long as the projective invariants are
bounded, the holonomy of the generators are bounded. (The term “holographic” will
be explained in our new paper [22]. This was spoken about in our talk in Melbourne,
May 18, 2009, and a pdf file of the presentation is available from the author).
Thus, it follows that J is the set given by projective invariants of the (3, 3, 3) boundary orbifolds satisfying some equations that the cross ratio of an edge are same from one
boundary orbifold to the other and that the products of Goldman σ-invariants equal
1 for some quadriples of Goldman σ-invariants. Therefore, J is a connected subset.
From this, we see that Def E (O) is connected. (We will explain more details later.)
Consider hol ◦ s : J → repE (π1 (O), PGL(4, R)). Actually, we find a lift s0 : J →
HomE (π1 (O), PGL(4, R) that is a continuous map.
We now show that the set D of convex irreducible elements of Def E (O) deformable
to hyperbolic ones are in SDef E,ce (O).
Since the ends of any convex real projective structures are always horospherical or
totally geodesic and lens-shaped by Proposition 4.20.
Let us take an irreducible element µ, deformable to a hyperbolic structure, and the
universal cover Ω of the corresponding convex real projective structure. We may assume
that Ω contains T1 by changing the developing maps if necessary. Next, Ω cannot be
reducible, i.e., Ω contain a totally geodesic subspace in the boundary preserved under
the holonomy group. Such a subspace cannot meet the interior of T1 since otherwise
the plane meets T1 but not some of the edges and hence cannot be invariant. Taking a
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maximal such subspace, we obtain a totally geodesic properly convex compact subset
P in the boundary. This means that there exists a straight segment in the boundary of
Ω∞ . However, our O always satisfies the no-edge condition. Take a maximal segment
and it must in a component U 0 of the closure of Ũ . Any maximal segment in the
subspace P meeting this is also in U 0 , and hence P is in U 0 . Since there are only unique
image of P under holonomy transformations by reducibility. This means that U 0 cannot
be sent to a disjoint component of the closure of Ũ , which contradict the fact that π1 (O)
is relatively hyperbolic and U 0 must have infinitely many images corresponding to the
number of cosets of π1 (O)/π1 (E) for the corresponding end E.
We show that D is closed in Def E (O). Since D is an open subset by Theorem , this
means that D = Def E (O) and so hence SDef E,ce (O) = Def E (O).
Suppose that we have a sequence of elements ji ∈ J, i = 1, 2, 3, . . . , so that we have
0
s (ji ) ∈ D and s0 (ji ) converges to an element h∞ of HomE (π1 (O), PGL(4, R)). Let Ωi
denote the developing image and Γi denote the image of the holonomy homomorphism.
We claim that Ωi is always a subset of the intersection of four half-spaces at vertices
of T1 and T2 a fixed adjacent tetrahedron. Then denote by Hji for j = 0, 1, 2, 3, 4, 5
denoteSthe supporting half spaces at vertices of T1 and T2 . Then we claim that
Ωi ⊂ Hji : This can be proved by induction. In fact, we can show that any finite
connected union of images of T1 , T2 under deck transformations is in an intersection
of supporting half-spaces at their vertices. Since there are five half-spaces containing
[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], and [2, 2, 2, −1], we can show easily that the
intersection is properly convex and in fact there is an upper bound to the length of
convex segments in it by π − 0 for a uniform 0 > 0 independent of i.
Let Ω∞ be the Hausdorff limit. Then Ω∞ is convex and further more properly
convex because of the length upper bound. Thus, Ω∞ is a properly convex subset with
nonempty interior containing T1o , and h∞ (π1 (O)) acts on it properly discontinuously.
The limit satisfies the no-edge condition: This follows since (Õ, Ũ ) being relatively
hyperbolic is quasi-isometry invariant. As the limit is also quasi-isometric, this follows.
Finally, Ω∞ cannot be reducible as can be proved similarly to above.
This proves that D is closed. Thus, we see that the closure of the image of s0 is in
the image of s0 .
Since the image of hol ◦ s : J → rep(π1 (O), PGL(4, R)) can be always conjugated
to be in the image of s0 , the image of hol ◦ s is also closed and consists of irreducible
representations. Since s is a homeomorphism, the image of hol is closed. As hol is an
open-map also, this completes the proof.
Since the image always lies in repiE,ce (π1 (O), PGL(4, R)), we see that our component
in repE (π1 (O), PGL(4, R)) is also a component of this space.
We remark that the above theorem can be generalized to orders ≥ 3. This will be
done in another paper [22].
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1.0
0.5
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Department of Mathematics, KAIST, Daejeon 305-701, South Korea
E-mail address: [email protected]
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