The convex real projective manifolds and orbifolds with radial ends
The convex real projective manifolds and orbifolds with
radial ends I: the openness of deformations (Preliminary)
Suhyoung Choi
Department of Mathematical Science
KAIST, Daejeon, South Korea
mathsci.kaist.ac.kr/∼schoi (Copies of my lectures are posted)
1/46
The convex real projective manifolds and orbifolds with radial ends
Abstract
I
A real projective orbifold is an n-dimensional orbifold modeled on RP n with the
group PGL(n + 1, R).
I
We concentrate on an orbifold with a compact codimension 0 submanifold whose
complement is a union of neighborhoods of ends, diffeomorphic to
(n − 1)-dimensional orbifolds times intervals.
2/46
The convex real projective manifolds and orbifolds with radial ends
Abstract
I
A real projective orbifold is an n-dimensional orbifold modeled on RP n with the
group PGL(n + 1, R).
I
We concentrate on an orbifold with a compact codimension 0 submanifold whose
complement is a union of neighborhoods of ends, diffeomorphic to
(n − 1)-dimensional orbifolds times intervals.
I
A real projective orbifold has radial ends if a neighborhood of each end is foliated
by projective geodesics concurrent to one another.
I
It is said to be convex if any path can be homotoped to a projective geodesic with
endpoints fixed.
2/46
The convex real projective manifolds and orbifolds with radial ends
Abstract
I
A real projective orbifold is an n-dimensional orbifold modeled on RP n with the
group PGL(n + 1, R).
I
We concentrate on an orbifold with a compact codimension 0 submanifold whose
complement is a union of neighborhoods of ends, diffeomorphic to
(n − 1)-dimensional orbifolds times intervals.
I
A real projective orbifold has radial ends if a neighborhood of each end is foliated
by projective geodesics concurrent to one another.
I
It is said to be convex if any path can be homotoped to a projective geodesic with
endpoints fixed.
I
A real projective structure on such an orbifold sometimes admits deformations to
inequivalent parameters of real projective structures, giving us a nontrivial
deformation space.
2/46
The convex real projective manifolds and orbifolds with radial ends
Abstract continued
I
We will prove the local homeomorphism between the deformation space of real
projective structures on such an orbifold with radial ends with various conditions
and the PGL(n + 1, R)-representation space of the fundamental group with
corresponding conditions.
I
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.
3/46
The convex real projective manifolds and orbifolds with radial ends
Abstract continued
I
We will prove the local homeomorphism between the deformation space of real
projective structures on such an orbifold with radial ends with various conditions
and the PGL(n + 1, R)-representation space of the fundamental group with
corresponding conditions.
I
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.
I
We will use a Hessian argument to show that a small deformation of a real
projective orbifold with ends 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.
I
We will also prove the closedness of the convex real projective structures on
orbifolds with irreducibilty condition.
3/46
The convex real projective manifolds and orbifolds with radial ends
Remarks
I
The understanding of the radial ends is not accomplished in this paper as this
forms an another subject.
I
We assume certain "natural" conditions to ends general enough and amenable to
study. These will be our base colonies.
I
We are trying to classify all possible types of ends (with Yves Benoist.) Even
simple examples are unclear.
4/46
The convex real projective manifolds and orbifolds with radial ends
Remarks
I
The understanding of the radial ends is not accomplished in this paper as this
forms an another subject.
I
We assume certain "natural" conditions to ends general enough and amenable to
study. These will be our base colonies.
I
We are trying to classify all possible types of ends (with Yves Benoist.) Even
simple examples are unclear.
I
This work is mostly theoretical, and we need more computable examples. What
might be the good conditions on the ends for computational purposes? (Benoist)
(For Coxeter orbifolds, see Choi and Marquis.) I gave computations talk in
Melbourne Univ. (in my homepage).
I
This work is still in progress. Consulting experts still..
4/46
The convex real projective manifolds and orbifolds with radial ends
Remarks
I
The understanding of the radial ends is not accomplished in this paper as this
forms an another subject.
I
We assume certain "natural" conditions to ends general enough and amenable to
study. These will be our base colonies.
I
We are trying to classify all possible types of ends (with Yves Benoist.) Even
simple examples are unclear.
I
This work is mostly theoretical, and we need more computable examples. What
might be the good conditions on the ends for computational purposes? (Benoist)
(For Coxeter orbifolds, see Choi and Marquis.) I gave computations talk in
Melbourne Univ. (in my homepage).
I
This work is still in progress. Consulting experts still..
I
Some of this might be known also by Cooper and Tillmann independently and
simultaneously: More computed examples for 3-manifolds admitting complete
hyperbolic structures.
4/46
The convex real projective manifolds and orbifolds with radial ends
Outline
Introduction
Orbifolds and real projective structures
Deformation spaces and holonomy maps
Classification of ends: rather restrictions on ends
Main results: Openness
Convexity and convex domains
The IPC-structures and relative hyperbolicity
The proof of the main results
Main Examples
Classification of the ends
5/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Orbifolds and real projective structures
Orbifolds
I
By an n-dimensional orbifold, we mean a Hausdorff second countable topological
space with a fine open cover {Ui , i ∈ I} with models (Ũi , Gi ) where Gi is a finite
group acting on the open subset Ũi of Rn and a map pi : Ũi → Ui inducing
homeomorphism Ũi /Gi → Ui where
I
for each i, j, x ∈ Ui ∩ Uj , there exists Uk with x ∈ Uk ⊂ Ui ∩ Uj .
I
given an inclusion Uj → Ui induces an equivariant map Ũj → Ũi with respect to
Gj → Gi .
6/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Orbifolds and real projective structures
Orbifolds
I
By an n-dimensional orbifold, we mean a Hausdorff second countable topological
space with a fine open cover {Ui , i ∈ I} with models (Ũi , Gi ) where Gi is a finite
group acting on the open subset Ũi of Rn and a map pi : Ũi → Ui inducing
homeomorphism Ũi /Gi → Ui where
I
for each i, j, x ∈ Ui ∩ Uj , there exists Uk with x ∈ Uk ⊂ Ui ∩ Uj .
I
given an inclusion Uj → Ui induces an equivariant map Ũj → Ũi with respect to
Gj → Gi .
I
A real projective structure on an orbifold is given by having charts from Ui s to
open subsets of RP n with transition maps in PGL(n + 1, R).
6/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Orbifolds and real projective structures
Orbifolds
I
By an n-dimensional orbifold, we mean a Hausdorff second countable topological
space with a fine open cover {Ui , i ∈ I} with models (Ũi , Gi ) where Gi is a finite
group acting on the open subset Ũi of Rn and a map pi : Ũi → Ui inducing
homeomorphism Ũi /Gi → Ui where
I
for each i, j, x ∈ Ui ∩ Uj , there exists Uk with x ∈ Uk ⊂ Ui ∩ Uj .
I
given an inclusion Uj → Ui induces an equivariant map Ũj → Ũi with respect to
Gj → Gi .
I
A real projective structure on an orbifold is given by having charts from Ui s to
open subsets of RP n with transition maps in PGL(n + 1, R).
I
A good orbifold: M/Γ where Γ is a discrete group with a properly discontinuous
order.
6/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Orbifolds and real projective structures
Real projective structures on orbifolds
I
I
Suppose that a discrete group Γ act on a manifold M properly discontinuously.
A real projective structure on M/Γ with simply connected M is given by an
immersion D : M → RP n equivariant with respect to a homomorphism
h : Γ → PGL(n + 1, R) where Γ is the fundamental group of M/Γ.
7/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Orbifolds and real projective structures
Real projective structures on orbifolds
I
I
Suppose that a discrete group Γ act on a manifold M properly discontinuously.
A real projective structure on M/Γ with simply connected M is given by an
immersion D : M → RP n equivariant with respect to a homomorphism
h : Γ → PGL(n + 1, R) where Γ is the fundamental group of M/Γ.
I
A real projective structure on M/Γ is convex if D(M) is a convex domain in an
affine subspace An ⊂ RP n . In this case, we will identify M with D(M) for a
particular choice of D and Γ with its image under h.
I
A properly convex domain is a convex domain that is a precompact domain in
some affine subspace. A convex domain is properly convex iff it does not contain
a complete real line.
7/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Orbifolds and real projective structures
Real projective structures on orbifolds
I
I
Suppose that a discrete group Γ act on a manifold M properly discontinuously.
A real projective structure on M/Γ with simply connected M is given by an
immersion D : M → RP n equivariant with respect to a homomorphism
h : Γ → PGL(n + 1, R) where Γ is the fundamental group of M/Γ.
I
A real projective structure on M/Γ is convex if D(M) is a convex domain in an
affine subspace An ⊂ RP n . In this case, we will identify M with D(M) for a
particular choice of D and Γ with its image under h.
I
A properly convex domain is a convex domain that is a precompact domain in
some affine subspace. A convex domain is properly convex iff it does not contain
a complete real line.
I
A real projective structure on M/Γ is properly convex if so is D(M).
7/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Orbifolds and real projective structures
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Figure: The developing images of 2-orbifolds
8/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Orbifolds and real projective structures
Motivations to study real projective structures
I
The study of lattices are very much established with many techniques.
I
Flexible geometric structures parametrize representations in many cases and they
do not correspond to lattice situations mostly.
9/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Orbifolds and real projective structures
Motivations to study real projective structures
I
The study of lattices are very much established with many techniques.
I
Flexible geometric structures parametrize representations in many cases and they
do not correspond to lattice situations mostly.
I
Real projective and conformal structures are often the most flexible
finite-dimensional types we can study. Other geometries are subgeometries.
I
Orbifolds with convex real projective structures have many properties of
CAT (0)-spaces and the theory is compatible with the geometric group theory. (no
angles... )
9/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Orbifolds and real projective structures
Motivations to study real projective structures
I
The study of lattices are very much established with many techniques.
I
Flexible geometric structures parametrize representations in many cases and they
do not correspond to lattice situations mostly.
I
Real projective and conformal structures are often the most flexible
finite-dimensional types we can study. Other geometries are subgeometries.
I
Orbifolds with convex real projective structures have many properties of
CAT (0)-spaces and the theory is compatible with the geometric group theory. (no
angles... )
I
There are "much" more orbifolds with real projective structures than homogeneous
Riemannian ones.
9/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Orbifolds and real projective structures
Motivations to study real projective structures on 3-orbifolds
I
Real projective structures on surfaces and 2-orbifolds are “understood". There is a
constructive classification from the convex decomposition theorem and the
annulus decomposition theorem.
I
There is a study by Cooper, Long, and Thistleswaite on the real projective
structures obtained by deforming hyperbolic 3-manifolds in the Hodgson-Weeks
census of 5000. About 5 percents are deformable.
10/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Orbifolds and real projective structures
Motivations to study real projective structures on 3-orbifolds
I
Real projective structures on surfaces and 2-orbifolds are “understood". There is a
constructive classification from the convex decomposition theorem and the
annulus decomposition theorem.
I
There is a study by Cooper, Long, and Thistleswaite on the real projective
structures obtained by deforming hyperbolic 3-manifolds in the Hodgson-Weeks
census of 5000. About 5 percents are deformable.
I
For the reflection 3-orbifolds, a study of orderable reflection orbifolds by Choi and
Marquis.
I
For more complicated reflection 3-orbifolds, Choi, Hodgson, and Lee made some
studies on hyperbolic cubes and dodecahedra. (Some deform as bending and as
nonbendings and some do not.) Ideal or hyperideal reflection 3-orbifolds without
order two deform always.
10/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Orbifolds and real projective structures
Motivations to study real projective structures on 3-orbifolds
I
Real projective structures on surfaces and 2-orbifolds are “understood". There is a
constructive classification from the convex decomposition theorem and the
annulus decomposition theorem.
I
There is a study by Cooper, Long, and Thistleswaite on the real projective
structures obtained by deforming hyperbolic 3-manifolds in the Hodgson-Weeks
census of 5000. About 5 percents are deformable.
I
For the reflection 3-orbifolds, a study of orderable reflection orbifolds by Choi and
Marquis.
I
For more complicated reflection 3-orbifolds, Choi, Hodgson, and Lee made some
studies on hyperbolic cubes and dodecahedra. (Some deform as bending and as
nonbendings and some do not.) Ideal or hyperideal reflection 3-orbifolds without
order two deform always.
I
Question (Cooper): Does every hyperbolic 3-orbifold deform up to finite covers?
I
How to make sense?
10/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Deformation spaces and holonomy maps
Deformation spaces of convex real projective structures
I
Given an orbifold S, a convex real projective structure is given by a diffeomorphism
f : S → Ω/Γ for a convex domain Ω in RP n and Γ a subgroup of PGL(n + 1, R).
I
This induces a diffeomorphism D : S̃ → Ω equivariant with respect to
h : π1 (S) → Γ.
11/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Deformation spaces and holonomy maps
Deformation spaces of convex real projective structures
I
Given an orbifold S, a convex real projective structure is given by a diffeomorphism
f : S → Ω/Γ for a convex domain Ω in RP n and Γ a subgroup of PGL(n + 1, R).
I
This induces a diffeomorphism D : S̃ → Ω equivariant with respect to
h : π1 (S) → Γ.
I
The deformation space CDef(S) of convex real projective structures is
{(D, h)}/ ∼ where (D, h) ∼ (D 0 , h0 ) if there is an isotopy f̃ : S̃ → S̃ such that
D 0 = D ◦ f̃ and h0 (f̃ g f̃ −1 ) = h(g) for each g ∈ π1 (S) or D 0 = k ◦ D and
h0 (·) = kh(·)k −1 for k ∈ PGL(n + 1, R).
11/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Deformation spaces and holonomy maps
Deformation spaces of convex real projective structures
I
Given an orbifold S, a convex real projective structure is given by a diffeomorphism
f : S → Ω/Γ for a convex domain Ω in RP n and Γ a subgroup of PGL(n + 1, R).
I
This induces a diffeomorphism D : S̃ → Ω equivariant with respect to
h : π1 (S) → Γ.
I
The deformation space CDef(S) of convex real projective structures is
{(D, h)}/ ∼ where (D, h) ∼ (D 0 , h0 ) if there is an isotopy f̃ : S̃ → S̃ such that
D 0 = D ◦ f̃ and h0 (f̃ g f̃ −1 ) = h(g) for each g ∈ π1 (S) or D 0 = k ◦ D and
h0 (·) = kh(·)k −1 for k ∈ PGL(n + 1, R).
I
Alternatively, CDef(S) = {f : S → Ω/Γ}/ ∼ where f ∼ g for f : S → Ω/Γ and
g : S → Ω0 /Γ0 if there exists a projective diffeomorphism k : Ω/Γ → Ω0 /Γ0 so that
k ◦ f is homotopic to g.
11/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Deformation spaces and holonomy maps
End orbifold
I
A real projective orbifold has radial ends if each end has an end neighborhood
foliated by concurrent geodesics for each chart ending at the common point of
concurrency.
I
Each end has a neighborhood diffeomorphic to a closed orbifold times an open
interval.
12/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Deformation spaces and holonomy maps
End orbifold
I
A real projective orbifold has radial ends if each end has an end neighborhood
foliated by concurrent geodesics for each chart ending at the common point of
concurrency.
I
Each end has a neighborhood diffeomorphic to a closed orbifold times an open
interval.
I
Given an end, there is an end orbifold associated with the end. The radial foliation
has a transversal real projective structure and hence the end orbifold has an
induced real projective structure of one dimension lower.
I
The end orbifold is convex if O is convex. If the end orbifold is properly convex,
then we say that the end is a transversely properly convex end.
12/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Deformation spaces and holonomy maps
The reason why we study orbifolds with ends
I
The orbifolds with ends might be more computable. For example, ideal or
hyperideal reflection hyperbolic 3-orbifolds with no edge order two have local
deformation space of dimension 6.
13/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Deformation spaces and holonomy maps
The reason why we study orbifolds with ends
I
The orbifolds with ends might be more computable. For example, ideal or
hyperideal reflection hyperbolic 3-orbifolds with no edge order two have local
deformation space of dimension 6.
I
The general type of ends are not understood.
I
The theory should be at least as complicated as that of Kleinian groups.
13/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Deformation spaces and holonomy maps
The reason why we study orbifolds with ends
I
The orbifolds with ends might be more computable. For example, ideal or
hyperideal reflection hyperbolic 3-orbifolds with no edge order two have local
deformation space of dimension 6.
I
The general type of ends are not understood.
I
The theory should be at least as complicated as that of Kleinian groups.
I
We (Jaejeong Lee, a student of Kapovich) are exploring some examples currently.
Also, Crampon and Marquis are studying these theoretically with horospherical
ends from the generalized Kleinian group perspectives.
I
Radial properly convex ones have some chance of classifications (with Benoist).
13/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Deformation spaces and holonomy maps
I
We define DefE (O) to be the subspace of Def(O) consisting of real projective
structures with radial ends.
I
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 .
14/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Deformation spaces and holonomy maps
I
We define DefE (O) to be the subspace of Def(O) consisting of real projective
structures with radial ends.
I
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 .
I
We define repE (π1 (O), PGL(n + 1, R)) to be the subspace of representations
where each end fundamental group has a nonzero common eigenvector.
I
The end fundamental group condition: If a representation of an end fixes a point of
RP n , then it fixes a unique one.
14/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Deformation spaces and holonomy maps
The hol map: The local homeomorphism property
Theorem A
Let O be a noncompact topologically tame n-orbifold. Suppose that O has the end
fundamental group conditions. Then the following map is a local homeomorphism:
hol : DefE (O) → repE (π1 (O), PGL(n + 1, R)).
Proof.
This follows as in the compact cases using the bump functions. However, we may need
the bump functions extending to the ends for radial ends.
15/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
I
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.
I
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.
I
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.
16/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
I
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.
I
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.
I
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.
I
A cone-over a lens-shaped domain A is a convex submanifold that contains a
lens-shaped domain A of the same dimension and
I
is a union of segments from a cone-point v 6∈ A to points of A,
I
the manifold boundary is one of the two boundary components of A, and
I
each maximal segment from v meets the two boundary components at unique points.
16/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
Figure: The universal covers of horospherical and lens shaped ends. The radial lines form
cone-structures.
17/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
I
A lens-cone is the union of the segments over a lens-shaped domain.
I
A lens is the lens-shaped domain A, not determined uniquely by the lens-cone
itself.
17/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
I
A lens-cone is the union of the segments over a lens-shaped domain.
I
A lens is the lens-shaped domain A, not determined uniquely by the lens-cone
itself.
I
I
A totally-geodesic 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.
17/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
I
A lens-cone is the union of the segments over a lens-shaped domain.
I
A lens is the lens-shaped domain A, not determined uniquely by the lens-cone
itself.
I
I
A totally-geodesic 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.
I
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 .
I
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 .
17/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
I
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.
I
An admissible group is a finite extension of a finite product of Zl × Γ1 × · · · × Γk
18/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
I
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.
I
An admissible group is a finite extension of a finite product of Zl × Γ1 × · · · × Γk
I
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.
I
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 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.
18/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
I
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.
I
An admissible group is a finite extension of a finite product of Zl × Γ1 × · · · × Γk
I
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.
I
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 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.
I
µ is a totally-geodesic representation if µ(π1 (E) acts on a cone-over a
totally-geodesic subdomain with a cone-point x.
I
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}.
I
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.
18/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
I
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) for a corresponding end orbifold E.
I
We require that the cone-point has to correspond to the end of the radial lines for
the given radial end.
19/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
I
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) for a corresponding end orbifold E.
I
We require that the cone-point has to correspond to the end of the radial lines for
the given radial end.
I
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.
I
l ≥ k follows from the result of Benoist.
19/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
I
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) for a corresponding end orbifold E.
I
We require that the cone-point has to correspond to the end of the radial lines for
the given radial end.
I
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.
I
l ≥ k follows from the result of Benoist.
I
We have k = 1 and l = 0 if and only if the end fundamental group is hyperbolic.
I
There are hyperbolic lens ends that are not totally geodesic. We can bend...
I
Lens and l ≥ k ≥ 1 imply totally geodesic
19/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
Convex end fundamental condition
I
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.
I
For example, the end has a 1-dimensional singularity following the radial line, this
is true.
20/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
Convex end fundamental condition
I
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.
I
For example, the end has a 1-dimensional singularity following the radial line, this
is true.
I
More generally if the fundamental group is virtually center free, this is true.
(Irreducibility essentially proves this..)
20/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
Convex end fundamental condition
I
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.
I
For example, the end has a 1-dimensional singularity following the radial line, this
is true.
I
More generally if the fundamental group is virtually center free, this is true.
(Irreducibility essentially proves this..)
I
For abelian fundamental groups, we need some singularity as above...
20/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
I
Let O be a tame n-orbifold where end fundamental groups are admissible.
I
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.)
21/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
I
Let O be a tame n-orbifold where end fundamental groups are admissible.
I
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.)
I
We define repiE,ce (π1 (O), PGL(n + 1, R)) as ...
I
We define DefiE,ce (O) to be the deformation space of real projective structures
with admissible ends and irreducible holonomy and define CDefE,ce (O) to be the
deformation space of irreducible properly convex-structures with admissible ends
(or IPC-structures).
21/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
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
I
In DefiE,ce (O), the subspace CDefE (O) of IPC-structures is open.
I
Suppose further that π1 (O) contains no notrivial nilpotent normal subgroup. The
deformation space CDefE,ce (O) of IPC-structures on O maps homeomorphic to a
component of repiE,ce (π1 (O), PGL(n + 1, R)).
22/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
Theorem C
I
Suppose that O is IPC. 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.
23/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
Theorem C
I
Suppose that O is IPC. 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.
I
We show that an IPC-orbifold O with admissible end is strictly IPC iff π1 (O) is
relatively hyperbolic with respect to its end fundamental groups using Bowditch
and Drutu-Sapir’s work.
23/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
Theorem C
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
I
π1 (O) is relatively hyperbolic with respect to its end fundamental groups.
24/46
The convex real projective manifolds and orbifolds with radial ends
Introduction
Classification of ends: rather restrictions on ends
Theorem C
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
I
π1 (O) is relatively hyperbolic with respect to its end fundamental groups.
I
In DefiE,ce (O), the subspace SDefiE (O) of strict IPC-structures with respect to the
ends is open.
I
The deformation space SDefE,ce (O) of strict IPC-structures on O with respect to
the ends maps homeomorphic to a component of
repiE,ce (π1 (O), PGL(n + 1, R)).
24/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
Convexity.
I
We begin by discussing the convexity:
Proposition
(Vey)
I
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 .
I
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 .
I
If a convex real projective orbifold with nonempty radial end is not properly convex, then
its holonomy is reducible.
25/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
Proposition
(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.
I
Every subgroup of finite index of Γ has a finite center.
I
Every subgroup of finite index of Γ has a trivial center.
I
Every subgroup of finite index of Γ is irreducible in PGL(n, R). That is, Γ is strongly
irreducible.
I
The Zariski closure of Γ is semisimple.
I
Γ does not contain a normal infinite nilpotent subgroup.
I
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.
26/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
I
Theorem
(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.
27/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
I
Theorem
(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.
I
Remark: This is the theorem we wish to generalize in the setting.
27/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
Facts on horospherical ends
Proposition
Let O be a topologically tame properly convex real projective n-orbifold with radial
ends.
I
For each horospherical end, the space of ray from the end point form a complete
affine space of dimension n − 1.
I
The only eigenvalues of g for an element of a horospherical end fundamental
group are 1 or complex numbers of absolute value 1.
I
An end point of a horospherical end cannot be on a segment in bdÕ.
I
For any compact set K inside a horospherical neighborhood, there exists a
horospherical ellipsoid neighborhood disjoint from K .
I
Let E be a complete end. Suppose that π1 (E) has holonomy with eigenvalues of
absolute value 1 only. Then E is horospherical.
Comment: Actually, I don’t need the eigenvalue condition for the final item. (Related to
the Auslander conjecture.)
28/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
I
If π1 (E) is hyperbolic, then E is lens. But if π1 (E) has many factors, this is unclear.
29/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
I
I
If π1 (E) is hyperbolic, then E is lens. But if π1 (E) has many factors, this is unclear.
Proposition
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.
29/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
I
I
If π1 (E) is hyperbolic, then E is lens. But if π1 (E) has many factors, this is unclear.
Proposition
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.
I
2
Let O be a 3-orbifold with the end orbifolds S3,3,3
. Then the orbifold has
admissible ends.
29/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The IPC-structures and relative hyperbolicity
Reminding Definitions Again
Definition
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.
30/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The IPC-structures and relative hyperbolicity
I
A Hilbert metric on an IPC-structure is defined as a distance metric given by cross
ratios. (We do not assume strictness here.)
I
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.
31/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The IPC-structures and relative hyperbolicity
I
A Hilbert metric on an IPC-structure is defined as a distance metric given by cross
ratios. (We do not assume strictness here.)
I
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.
I
This gives us a well-defined Finsler metric.
I
Given an IPC-structure on O, there is a Hilbert metric dH on Õ and hence on Õ.
This induces a metric on O.
31/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The IPC-structures and relative hyperbolicity
I
We will use Bowditch’s result to show
Theorem (D)
Let O be a topologically tame strictly IPC-orbifold with radial ends and has no essential
annuli or tori. Then π1 (O) is relatively hyperbolic with respect to the end groups
π1 (E1 ), ..., π1 (Ek ).
32/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The IPC-structures and relative hyperbolicity
I
We will use Bowditch’s result to show
Theorem (D)
Let O be a topologically tame strictly IPC-orbifold with radial ends and has no essential
annuli or tori. Then π1 (O) is relatively hyperbolic with respect to the end groups
π1 (E1 ), ..., π1 (Ek ).
I
Fact: If π1 (El ), .., π1 (Ek ) are hyperbolic for some 0 ≤ l < k , then π1 (O) is
relatively hyperbolic with respect to π1 (E1 ), . . . , π1 (El−1 ). (Drutu)
32/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The IPC-structures and relative hyperbolicity
I
Proof: We denote this quotient space bdÕ1 / ∼ by B.
I
We will use Theorem 0.1. of Yaman [6]: We show that π1 (O) acts on the set B as
a geometrically finite convergence group.
33/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The IPC-structures and relative hyperbolicity
I
Proof: We denote this quotient space bdÕ1 / ∼ by B.
I
We will use Theorem 0.1. of Yaman [6]: We show that π1 (O) acts on the set B as
a geometrically finite convergence group.
I
The group acts properly discontinuously on the set of triples in B.
33/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The IPC-structures and relative hyperbolicity
I
Proof: We denote this quotient space bdÕ1 / ∼ by B.
I
We will use Theorem 0.1. of Yaman [6]: We show that π1 (O) acts on the set B as
a geometrically finite convergence group.
I
I
The group acts properly discontinuously on the set of triples in B.
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. The groups of form Γx are the only parabolic subgroups. Also,
(B − {x})/Γx is easily seen to be homeomorphic to the end orbifold and
therefore, compact. Hence, each Γx is a bounded parabolic subgroup.
33/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The IPC-structures and relative hyperbolicity
I
Proof continued: Let p be a point that is not a horospherical endpoint or a
singleton corresponding an lens-shaped end. We show that p is a conical limit
point.
34/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The IPC-structures and relative hyperbolicity
I
Proof continued: Let p be a point that is not a horospherical endpoint or a
singleton corresponding an lens-shaped end. We show that p is a conical limit
point.
I
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.
34/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The IPC-structures and relative hyperbolicity
I
Proof continued: Let p be a point that is not a horospherical endpoint or a
singleton corresponding an lens-shaped end. We show that p is a conical limit
point.
I
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.
p'
l
q'
m
p
q
Figure: A shortest geodesic m to a geodesic l.
34/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The IPC-structures and relative hyperbolicity
Converse
We will prove the partial converse to the above Theorem D:
Theorem (E)
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, ..., k for 0 ≤ m ≤ k .
I
Assume that O is IPC. Then O is strictly IPC.
I
Let O1 be obtained by removing the concave neighborhoods of hyperbolic ends.
Then if O is IPC, then O1 is strictly IPC.
(Note: This improves the theorem in the paper.)
35/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The IPC-structures and relative hyperbolicity
Proof.
Suppose not. We obtain a triangle T with ∂T in ∂ Õ1 .
Lemma
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.
Uses asymptotic cone in Drutu-Sapir’s work.
36/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The proof of the main results
I
Then we define SDefE,ce (O) to be the subspace of DefiE,ce (O) consisting of strict
IPC-structures. Also, we have SDefRP n ,E,ce (O) ⊂ DefiRP n ,E,ce (O).
37/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The proof of the main results
I
Then we define SDefE,ce (O) to be the subspace of DefiE,ce (O) consisting of strict
IPC-structures. Also, we have SDefRP n ,E,ce (O) ⊂ DefiRP n ,E,ce (O).
I
Theorem
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 DefiE,ce (O), the subspace CDefE (O, ce) of IPC-structures is open, and so
is SDefE (O, ce).
37/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The proof of the main results
Corollary
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.
hol : CDefE,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 so is
hol : SDefE,ce (O) → repE,ce (π1 (O), PGL(n + 1, R)).
38/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The proof of the main results
Proof
I
We increase the end neighborhoods to approximate Õ.
I
In the affine suspension cone V and its dual cone V ∗ , we find Koszul-Vinberg
function
Z
fV ∗ (x) =
e−φ(x) dφ
(1)
V∗
39/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The proof of the main results
Proof
I
We increase the end neighborhoods to approximate Õ.
I
In the affine suspension cone V and its dual cone V ∗ , we find Koszul-Vinberg
function
Z
fV ∗ (x) =
I
e−φ(x) dφ
(1)
V∗
Then, we deform O. We patch together the deformed functions to obtain a
function with positive definite Hessian. This implies the convexity.
I
This proves the openess of IPC-structures.
39/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The proof of the main results
Proof
I
We increase the end neighborhoods to approximate Õ.
I
In the affine suspension cone V and its dual cone V ∗ , we find Koszul-Vinberg
function
Z
fV ∗ (x) =
I
e−φ(x) dφ
(1)
V∗
Then, we deform O. We patch together the deformed functions to obtain a
function with positive definite Hessian. This implies the convexity.
I
This proves the openess of IPC-structures.
I
To show the openness of strict IPC-structures, we need the fact that small
deformation of O preserves relative hyperbolicity and hence the strictness of the
IPC-structures.
39/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The proof of the main results
Closedness
I
Theorem
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:
I
The deformation space CDefE,ce (O) of IPC-structures on O maps homeomorphic to the
union of components of repiE,ce (π1 (O), PGL(n + 1, R)).
40/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The proof of the main results
Closedness
I
Theorem
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:
I
The deformation space CDefE,ce (O) of IPC-structures on O maps homeomorphic to the
I
Suppose also that O has no essential homotopy annulus or torus. Similarly, the same
union of components of repiE,ce (π1 (O), PGL(n + 1, R)).
can be said for SDefE,ce (O) of strict IPC-structures on O.
40/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
The proof of the main results
Closedness
I
Theorem
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:
I
The deformation space CDefE,ce (O) of IPC-structures on O maps homeomorphic to the
I
Suppose also that O has no essential homotopy annulus or torus. Similarly, the same
union of components of repiE,ce (π1 (O), PGL(n + 1, R)).
can be said for SDefE,ce (O) of strict IPC-structures on O.
I
We can drop the irreducibility in the representations space:
Corollary
Assume as above: 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 repE,ce (π1 (O), PGL(n + 1, R)).
40/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
Main Examples
S. Tillman’s example
I
There is a census of small hyperbolic orbifolds with graph-singularity [4]. (See the
paper by D. Heard, C. Hodgson, B. Martelli, and C. Petronio)
I
There is a complete hyperbolic structure on the orbifold based on S3 with handcuff
singularity with two points removed. The singularity orders are three.
41/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
Main Examples
S. Tillman’s example
I
There is a census of small hyperbolic orbifolds with graph-singularity [4]. (See the
paper by D. Heard, C. Hodgson, B. Martelli, and C. Petronio)
I
There is a complete hyperbolic structure on the orbifold based on S3 with handcuff
singularity with two points removed. The singularity orders are three.
I
There is a one-parameter space of deformations of the structures to real
projective structures by simple matrix computations.
I
These are all stricly IPC by our theory.
41/46
The convex real projective manifolds and orbifolds with radial ends
Convexity and convex domains
Main Examples
Main examples
Theorem
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 p ≥ 3. Then we have
DefE (O) = SDefE,ce (O).
(2)
and hol maps DefE (O) as an onto-map a component of representations
repE (π1 (O), PGL(4, R))
which is also a component of repiE,ce (π1 (O), PGL(4, R)).
42/46
The convex real projective manifolds and orbifolds with radial ends
Classification of the ends
The end classification
Let E be an end orbifold. Can the end be classified?
I
We consider the case when E is not affine nor properly convex.
I
Let E be a closed and convex but not properly convex real projective
(n − 1)-orbifold arising as an end of a IPC-orbifold of dimension n. The universal
cover Ẽ of E is foliated by affine subspaces Rm with common boundary RP m−1 .
There is a projection p : Ẽ → K where K is a properly convex open domain in
RP n−m−1 .
43/46
The convex real projective manifolds and orbifolds with radial ends
Classification of the ends
The end classification
Let E be an end orbifold. Can the end be classified?
I
We consider the case when E is not affine nor properly convex.
I
Let E be a closed and convex but not properly convex real projective
(n − 1)-orbifold arising as an end of a IPC-orbifold of dimension n. The universal
cover Ẽ of E is foliated by affine subspaces Rm with common boundary RP m−1 .
There is a projection p : Ẽ → K where K is a properly convex open domain in
RP n−m−1 .
I
Let Γ be a subgroup of PGL(n, R) and the deck transformation group of Ẽ. Then
there is an exact sequence
1 → ker → Γ → Γ(K ) → 1
where Γ(K ) acts on K divisibly or sweep K up.
I
We can show that there is a map Γ → Γf ⊂ PGL(m − 1) by restricting the maps to
RP m−1 . Then we can show that the image Γf has only eigenvalues of absolute
value 1 only. Thus Γf is virtually Solvable as shown by Fried mainly. (I think that
this is true.)
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The convex real projective manifolds and orbifolds with radial ends
Classification of the ends
The end classification
Let E be an end orbifold. Can the end be classified?
I
We consider the case when E is not affine nor properly convex.
I
Let E be a closed and convex but not properly convex real projective
(n − 1)-orbifold arising as an end of a IPC-orbifold of dimension n. The universal
cover Ẽ of E is foliated by affine subspaces Rm with common boundary RP m−1 .
There is a projection p : Ẽ → K where K is a properly convex open domain in
RP n−m−1 .
I
Let Γ be a subgroup of PGL(n, R) and the deck transformation group of Ẽ. Then
there is an exact sequence
1 → ker → Γ → Γ(K ) → 1
where Γ(K ) acts on K divisibly or sweep K up.
I
We can show that there is a map Γ → Γf ⊂ PGL(m − 1) by restricting the maps to
RP m−1 . Then we can show that the image Γf has only eigenvalues of absolute
value 1 only. Thus Γf is virtually Solvable as shown by Fried mainly. (I think that
this is true.)
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The convex real projective manifolds and orbifolds with radial ends
Classification of the ends
I
The space of representations ρ of π1 (E) for an n − 1-dimensional end E to
PGL(n − 1) can lift to ones in PGL(n) which fix a point.
I
Hence, the space of representation is Hρ1 (π1 (E), Rn−1,∗ ) × H 1 (π1 (E), R).
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The convex real projective manifolds and orbifolds with radial ends
Classification of the ends
I
The space of representations ρ of π1 (E) for an n − 1-dimensional end E to
PGL(n − 1) can lift to ones in PGL(n) which fix a point.
I
Hence, the space of representation is Hρ1 (π1 (E), Rn−1,∗ ) × H 1 (π1 (E), R).
I
We are still investigating which parts are relevant to our study.
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The convex real projective manifolds and orbifolds with radial ends
Classification of the ends
Horospherical or lens-shaped restrictions
The restriction of horospherical or lens be dropped?
I
We restrict to a properly convex end.
I
The end with a virtually abelian fundamental group.
I
We can find example where the lens-shaped condition is not true.
I
Call these examples properly convex abelian ends.
I
The join of this with a totally geodesic end give examples that are not lens-shaped.
I
These should be allowed.?
I
Are these all?
The picture we wish for
Thus, the end should be the join of properly convex abelian ends, totally geodesic
hyperbolic joined ends, and horospherical ends. or just hyperbolic lens-shaped end or
a horospherical end. The concept of join should be set theoretic and the representation
need not split...
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The convex real projective manifolds and orbifolds with radial ends
Classification of the ends
C. Drutu,
Relatively hyperbolic groups: Geometry and quasi-isometric invariance,
Comment. Math. Helv. 84 (2005), 503–546
C. Drutu and M. Sapir,
Tree-graded spaces and asymptotic cones of groups,
Topology 44 (2005), no. 5, 959–1058.
W. Goldman.
Geometric structures on manifolds and varieties of representations.
Contemp. Math., 74:169–198, 1988.
D. Heard, C. Hodgson, B. Martelli, and C. Petronio,
Hyperbolic graphs of small complexity,
Experiment. Math. 19 (2010), no. 2, 211–236
D. Osin and M. Sapir,
An appendix to "Tree-graded spaces and asymptotic cones of groups
Topology, 44 (2005), no. 5, 959–1058.
A. Yaman
A topological characterisation of relatively hyperbolic groups,
J. reine ange. Math. 566 (2004), 41 – 89.
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