LIMIT DIRECTIONS FOR LORENTZIAN COXETER SYSTEMS
arXiv:1403.1502v1 [math.GR] 6 Mar 2014
HAO CHEN† AND JEAN-PHILIPPE LABBɇ
Abstract. Every Coxeter group admits a geometric representation as a group generated by
reflections in a real vector space. In the projective representation space, limit directions are
limits of injective sequences in the orbit of some base point. Limit roots are limit directions
that can be obtained starting from simple roots. In this article, we study the limit directions
arising from any point when the representation space is a Lorentz space. In particular, we
characterize the light-like limit directions using eigenvectors of infinite-order elements. This
provides a spectral perspective on limit roots, allowing for efficient computations. Moreover, we
describe the space-like limit directions in terms of the projective Coxeter arrangement.
1. Introduction
Every Coxeter system has a linear representation as a reflection group acting on a real vector
space endowed with a canonical bilinear form, see [Bou68] or [Hum92]. The unit basis vectors of the
representation space are called simple roots. The set of vectors in the orbits of simple roots under
the action of the Coxeter group are called roots. The set of roots forms the root system associated
to the Coxeter system. Every root corresponds to a unique orthogonal reflecting hyperplane. The
set of reflecting hyperplanes is called the Coxeter arrangement.
For infinite Coxeter groups, Vinberg introduced more general representations using different
bilinear forms for the representation space [Vin71]. We use the notation (W, S)B to denote a
Coxeter system (W, S) associated with a matrix B that determines the bilinear form used to
represent (W, S). We refer to (W, S)B as a geometric Coxeter system. When the bilinear form has
signature (n − 1, 1), where n is the rank of (W, S), the representation space is a Lorentz space. In
this case, we say that (W, S)B is a Lorentzian Coxeter system. The action of the Coxeter group on
Lorentz space induces an action on hyperbolic space. Coxeter groups acting on hyperbolic space
with compact or finite volume fundamental domains are well studied; see for instance [Vin71]
[Hum92, Sections 6.8-9] [Rat06, Chapter 7] and [AB08, Sections 10.3-4].
Let (W, S)B be a geometric Coxeter system. The Coxeter group W acts linearly on the representation space V , therefore one may also consider its action on the corresponding projective
space PV . A point of PV is a limit direction of (W, S)B if it is the limit of an injective sequence
b0 ∈ PV . When the base point is a simple root, then the
of points in the orbit of some base point x
limit direction is called a limit root.
The notion of limit roots was introduced and studied in [HLR14]. Properties of limit roots for
infinite Coxeter systems were investigated in a series of papers: Limit roots lie on the isotropic cone
of the bilinear form associated to the representation space [HLR14, Theorem 2.7]. The convex cone
spanned by limit roots is studied in [Dye13, Theorem 5.4] as the imaginary cone. The relations
between limit roots and the imaginary cone are further investigated in [DHR13]. For irreducible
Coxeter systems, every limit root can be obtained from any root or from any limit root [DHR13,
Theorem 3.1(b,c)].
When the Coxeter group acts on a Lorentz space, every limit root can be obtained from any
time-like vector [HPR13, Theorem 3.3]. This result is obtained by interpreting the Coxeter group
2010 Mathematics Subject Classification. Primary 20F55, 37B05; Secondary 22E43, 52C35.
Key words and phrases. Coxeter groups, Lorentz space, limit set, Coxeter arrangement, infinite root systems,
fractal.
† supported by the Deutsche Forschungsgemeinschaft within the Research Training Group “Methods for Discrete
Structures” (GRK 1408).
‡ supported by a FQRNT Doctoral scholarship and SFB Transregio “Discretization in Geometry and Dynamics”
(TRR 109).
1
2
H. CHEN AND J.-P. LABBÉ
as a Kleinian group acting on the hyperbolic space. Furthermore, every limit root can be obtained
from weights [CL14, Theorem 3.4]. Theorem 2.5 of the present paper implies that limit directions
arising from light-like directions are limit roots.
Theorem 2.3 below summarizes known results on base points whose orbits accumulate at limit
roots under the action of the Coxeter group. These results motivate the investigation of limit
b0 ∈ PV . In this paper, we initiate
directions of Coxeter groups arising from any base point x
this examination when the representation space V is a Lorentz space. This approach includes the
action of Coxeter groups on hyperbolic spaces, roots and weights as special cases. Interestingly,
limit directions reveal behaviors that did not get noticed previously.
Based on the classification of Lorentzian transformations according to their eigenvalues, our
first main result introduces a spectral perspective for limit roots involving infinite-order elements
of the group.
Theorem 1.1. Let E∞ be the set of directions of non-unimodular eigenvectors for infinite-order
elements of a Lorentzian Coxeter system (W, S)B . The set of limit roots EΦ of (W, S)B is the
closure of E∞ , that is
EΦ = E∞ .
This spectral description provides a natural way to directly compute limit roots from the Coxeter group and its geometric action through eigenspaces of infinite-order elements. We have implemented a package in Sage [S+ 14] to do such computations; the package will be made available
in Sage during the Sage days 57 in Cernay [LR14]. In the previous articles [HLR14] [DHR13]
[HPR13] [CL14], figures showed partial approximations of the limit roots using roots or weights.
In contrast, Figure 1 presents two examples from our computations that reveal the full picture.
5
−1.05
−1.05
(a) Some 30080 limit roots of the geometric Coxeter system with the shown graph. They are
obtained from infinite-order elements of length
3 and 4 and their conjugates with elements of
length 1 to 9.
5
∞
(b) Some 28019 limit roots of the geometric
Coxeter system with the shown graph. They are
obtained from infinite-order elements of length
2, 3 and 4 and their conjugates with elements
of length 1 to 5.
Figure 1. Limit roots of two geometric Coxeter systems of rank 4 seen in the
affine space spanned by the simple roots.
In the hope of further generalizations to non-Lorentzian Coxeter systems, our proof of Theorem 1.1 tries to rely minimally on the hyperbolic geometric interpretation of [HPR13]. For
Lorentzian Coxeter systems, the set of limit roots equals the limit set of the Coxeter group seen as
a Kleinian group. Moreover, limit roots are indeed the only light-like limit directions, as we show
LIMIT DIRECTIONS
DIRECTIONS FOR
FOR LORENTZIAN
LORENTZIAN COXETER
COXETER SYSTEMS
SYSTEMS
LIMIT
33
in Theorem
Theorem 2.5,
2.5, which
which isis aa key
key to
to the
the proof
proof of
of Theorem
Theorem 1.1.
1.1. However,
However, for
for non-Lorentzian
non-Lorentzian Coxeter
Coxeter
in
systems, there
there may
may be
be isotropic
isotropic limit
limit directions
directions that
that are
are not
not limit
limit roots,
roots, as
as shown
shown in
in Example
Example 3.11.
3.11.
systems,
Furthermore, while
while proving
proving Theorem
Theorem 1.1,
1.1, we
we also
also observed
observed space-like
space-like limit
limit directions.
directions. Our
Our
Furthermore,
second main
main result
result describes
describes the
the set
set of
of limit
limit directions
directions for
for Lorentzian
Lorentzian Coxeter
Coxeter systems
systems in
in terms
terms of
of
second
the projective
projective Coxeter
Coxeter arrangement,
arrangement, i.e.
i.e. the
the infinite
infinite arrangement
arrangement of
of reflecting
reflecting hyperplanes
hyperplanes in
in the
the
the
projective representation
representation space.
space.
projective
Theorem
Theorem 1.2.
1.2. Let
Let E
EVV be
be the
the set
set of
of limit
limit directions
directions of
of aa Lorentzian
Lorentzian Coxeter
Coxeter system
system (W,
(W,S)
S)BB and
and
L
Lhyp
be the
the union
union of
of codimension-2
codimension-2 space-like
space-like intersections
intersections of
of the
the projective
projective Coxeter
Coxeter arrangement
arrangement
hyp be
associated
associated to
to (W,
(W,S)
S)BB.. The
The set
set of
of limit
limit directions
directions E
EVV is
is “sandwiched”
“sandwiched” between
between L
Lhyp
and its
its closure,
closure,
hyp and
that
that is
is
L
⊂ E ⊆ L hyp..
Lhyp
hyp ⊂ EVV ⊆ Lhyp
γ
α
β
Figure 2. Some reflecting hyperplanes in the projective Coxeter arrangement of
Figure
2. Some
reflecting
hyperplanes
theassociated
projectivebilinear
Coxeter
arrangement
of
the universal
Coxeter
group
of rank 3. in
The
form
has cij = 1.1
the
universal
group of rank
3. Theintersections
associated bilinear
formwith
has cdots.
ij = 1.1
whenever
i �=Coxeter
j. Codimension-2
space-like
are marked
By
whenever
6= j.the
Codimension-2
space-like
are marked
dots. By
Theorem i1.2,
intersections in
Lhyp areintersections
limit directions.
The sixwith
intersections
Theorem
1.2, diamonds
the intersections
in Lhyp are limit directions. The six intersections
marked with
are weights.
marked with diamonds are weights.
The hyperplane arrangement Lhyp involved in Theorem 1.2 is infinite and not discrete. Figure 2
The hyperplane arrangement Lhyp
involved in Theorem 1.2 is infinite and not discrete. Figure 2
shows part of the Coxeter arrangement
of a universal Coxeter group and some intersections in Lhyp .
shows part of the Coxeter arrangement of a universal Coxeter group and some intersections in Lhyp
.
While Theorem 1.2 has a combinatorial flavour, a stronger relation holds. Let Uhyp be the set
While Theorem 1.2 has a combinatorial flavour, a stronger relation holds. Let Uhyp be the set
of space-like unimodular eigendirections of infinite-order elements. Then the set of limit directions
of space-like unimodular eigendirections of infinite-order elements. Then the set of limit directions
satisfies
satisfies
Uhyp
�E
EΦΦ ⊆
⊆E
EVV ⊆
⊆U
Uhyp
hyp t
hyp..
U
Notably, space-like
space-like weights
weights are
are all
all limit
limit directions,
directions, as
as marked
marked by
by diamonds
diamonds in
in Figure
Figure 22 and
and 12.
12.
Notably,
Moreover, roots
roots may
may also
also be
be limit
limit directions.
directions. In
In Section
Section 4.4,
4.4, we
we discuss
discuss the
the possibility
possibility of
of equalities
equalities
Moreover,
on either
either side,
side, and
and point
point out
out certain
certain unexplained
unexplained linear
linear dependencies
dependencies among
among limit
limit directions
directions in
in
on
Uhyp
see Figure
Figure 12.
12.
U
hyp,, see
After we
we obtained
obtained the
the main
main results,
results, we
we noticed
noticed potential
potential relations
relations between
between limit
limit roots
roots and
and
After
two conjectures:
conjectures: aa conjecture
conjecture of
of Lam
Lam and
and Pylyavskyy,
Pylyavskyy, stating
stating that
that the
the limit
limit weak
weak order
order isis aa
two
lattice [LP13,
[LP13, Conjecture
Conjecture 10.3],
10.3], and
and aa conjecture
conjecture of
of Dyer,
Dyer, stating
stating that
that the
the extended
extended weak
weak order
order
lattice
complete ortholattice
ortholattice [Dye11,
[Dye11, Conjecture
Conjecture 2.5].
2.5]. The
The limit
limit weak
weak order
order extends
extends the
the usual
usual weak
weak
isis aa complete
order to
to infinite
infinite reduced
reduced words,
words, whereas
whereas the
the extended
extended weak
weak order
order extends
extends the
the usual
usual weak
weak order
order to
to
order
infinite biclosed
biclosed sets.
sets. In
In Section
Section 2.4,
2.4, we
we relate
relate limit
limit roots
roots to
to the
the study
study of
of infinite
infinite reduced
reduced words
words
infinite
4
H. CHEN AND J.-P. LABBÉ
and infinite biclosed sets for Lorentzian Coxeter systems. The relations between limit weak order
and extended weak order in general deserve to be explored in more detail.
The present paper is organized as follows. In Section 2, we recall the geometric representations
of Coxeter systems, define the notion of limit directions, and review some results on limit roots.
Then we prove that limit roots are the only light-like limit directions for Lorentzian Coxeter
systems, and study the relation between infinite reduced words and limit roots. In Section 3, we
recall spectral properties of Lorentz transformations, and prove Theorem 1.1. In the last part
of Section 3, we give an example of non-Lorentzian Coxeter system, for which some isotropic
limit directions are not limit roots. In Section 4, we define and study the projective Coxeter
arrangement, and prove Theorem 1.2. Finally, some open problems are discussed.
Acknowledgement. The authors would like to thank Christophe Hohlweg, Ivan Izmestiev and
Vivien Ripoll for helpful discussions. The second author is grateful to Christophe Hohlweg for
introducing him to infinite root systems during Summer 2010 and to Sébastien Labbé and Vivien
Ripoll for their help in the implementation of the Sage package.
2. Limit roots of Lorentzian Coxeter systems
2.1. Lorentzian Coxeter systems. A n-dimensional Lorentz space (V, B) is a vector space V
associated with a bilinear form B of signature (n − 1, 1). In a Lorentz space, a vector x is spacelike (resp. time-like, light-like) if B(x, x) is positive (resp. negative, zero). The set of light-like
vectors Q = {x ∈ V | B(x, x) = 0} forms a cone called the light cone. The following proposition
characterizes the totally-isotropic subspaces of Lorentz spaces.
Proposition 2.1 ([Cec08, Theorem 2.3]). Let (V, B) be a Lorentz space and x, y ∈ Q be two
light-like vectors. Then B(x, y) = 0 if and only if x = cy for some c ∈ R.
A linear transformation on V that preserves the bilinear form B is called a Lorentz transformation. The group of Lorentz transformations is called Lorentz group and noted OB (V ).
Let (W, S) be a finitely generated Coxeter system, where S is a finite set of generators and
the Coxeter group W is generated with the relations (st)mst = e, where s, t ∈ S, mss = 1
and mst = mts ≥ 2 or = ∞ if s 6= t. The cardinality |S| = n is the rank of the Coxeter
system (W, S). For an element w ∈ W , the length `(w) of w is the smallest natural number k such
that w = s1 s2 . . . sk for si ∈ S. We refer the readers to [Bou68, Hum92] for more details. We
associate a matrix B to (W, S) as follows:
(
− cos(π/mst ) if mst < ∞;
Bst =
−cst
if mst = ∞,
for s, t ∈ S, where cst are chosen arbitrarily with cst = cts ≥ 1. The Coxeter system (W, S)
associated with the matrix B is called a geometric Coxeter system and is noted by (W, S)B .
Let V be a real vector space of dimension n, equipped with a basis ∆ = {αs }s∈S . The matrix B
defines a bilinear form B on V by B(αs , αt ) = αs| Bαt , for s, t ∈ S. For a vector α ∈ V such that
B(α, α) 6= 0, we define the reflection σα
(1)
σα (x) := x − 2
B(x, α)
α
B(α, α)
for all x ∈ V.
The homomorphism ρ : W → GL(V ) sending s to σαs is a faithful geometric representation of the
Coxeter group W as a discrete group of Lorentz transformations. We refer the readers to [Kra09,
Chapter 1] and [HLR14, Section 1] for more details. In the following, we will write w(x) in place
of ρ(w)(x).
A geometric Coxeter system (W, S)B is of finite type if B is positive-definite. In this case, W
is a finite group. A geometric Coxeter system (W, S)B is of affine type if the matrix B is positive
semi-definite but not definite. In either case, the group W can be represented as a reflection group
in Euclidean space. If the matrix B has signature (n − 1, 1), the pair (V, B) is a n-dimensional
Lorentz space, and W acts on V as a discrete subgroup of the Lorentz group OB (V ). In this case,
LIMIT DIRECTIONS FOR LORENTZIAN COXETER SYSTEMS
5
we say that the geometric Coxeter system (W, S)B is Lorentzian and, by abuse of language, that W
is a Lorentzian Coxeter group. See [CL14, Remark 2.2] for further discussions about terminologies.
In the spirit of [CL14], we pass to the projective space PV , i.e. the topological space of 1b ∈ PV denote the line passing
dimensional subspaces of V . For a non-zero vector x ∈ V \ {0}, let x
through x and the origin. The group action of W on V by reflection induces a projective action
[ for w ∈ W and x ∈ V . For a set X ⊂ V , the corresponding projective
b = w(x),
of W on PV as w · x
b := {b
b
set is X
x ∈ PV | x ∈ X}. The projective light cone is denoted by Q.
Let h(x) denote the sum of the coordinates of x in the basis ∆, and call it the height of the
vector x. We say that x is future-directed (resp. past-directed) if h(x) is positive (resp. negative).
The hyperplane {x ∈ V | h(x) = 1} is the affine subspace aff(∆) spanned by the basis of V .
It is useful to identify the projective space PV with the affine subspace aff(∆) with a projective
b is identified with the vector
hyperplane added at infinity. For a vector x ∈ V , if h(x) 6= 0, x
x/h(x) ∈ aff(∆).
b is identified to a point on the projective hyperplane at
Otherwise, if h(x) = 0, the direction x
b is identified with the
infinity. For a basis vector α ∈ ∆, we have α
b = α. In fact, if h(x) 6= 0, x
intersection of aff(∆) with the straight line passing through x and the origin. The projective light
b is projectively equivalent to a sphere, see for instance [DHR13, Proposition 4.13]. The
cone Q
affine subspace aff(∆) is practical for visualization and geometric intuitions.
Given a topological space X and a subset Y ⊆ X, a point x ∈ X is an accumulation point of Y
if every neighborhood of x contains a point of Y different from x. Let G be a group acting on X,
then a point x ∈ X is a limit point of G if x is an accumulation point of the orbit G(x0 ) for some
base point x0 ∈ X. Alternatively, x is a limit point of G, if there is a base point x0 ∈ X and a
sequence of elements (gk )k∈N ∈ G such that the sequence of points (gk (x0 ))k∈N is injective and
converges to x as k → ∞. In this case, we say that x is a limit point of G acting on X arising
from the base point x0 through the sequence (gk )k∈N . We now define the main object of study of
the present paper.
Definition 2.2. Limit directions of a geometric Coxeter system (W, S)B are limit points of W
acting on PV . The set of limit directions is denoted by EV . In other words,
b0 )i∈N
EV = {b
x ∈ PV | there is a vector x0 ∈ V and an injective sequence (wi · x
b0 such that lim wi · x
b0 = x
b}.
in the orbit W · x
i→∞
2.2. Limit roots. We call the basis vectors in ∆ the simple roots. Let Φ = W (∆) be the orbit of ∆
under the action of W , then the vectors in Φ are called roots. The pair (Φ, ∆) is a based root system.
The roots Φ are partitioned into positive roots Φ+ = cone(∆) ∩ Φ and negative roots Φ− = −Φ+ .
The depth of a positive root γ ∈ Φ+ is the smallest integer k such that γ = s1 s2 . . . sk−1 (α), for
si ∈ S and α ∈ ∆.
Let V ∗ be the dual vector space of V with dual basis ∆∗ . If the bilinear form B is non-singular,
which is the case for Lorentz spaces, V ∗ can be identified with V , and ∆∗ = {ωs }s∈S can be
identified with a set of vectors in V such that
(2)
B(αs , ωt ) = δst ,
where δst is the Kronecker delta function. Vectors in ∆∗ are called fundamental weights, and
vectors in the orbit
[
Ω := W (∆∗ ) =
W (ω)
ω∈∆∗
are called weights. See [Bou68, Chapter VI, Section 1.10] and [Max82, Section 1] for more details
about the notion of weights.
Limit roots were introduced in [HLR14] to study infinite root systems arising from infinite
b in PV . In other
Coxeter groups. They are the accumulation points of the projective roots Φ
words, the set of limit roots is defined as follows
b}.
EΦ = {b
x ∈ PV | there is an injective sequence (γi )i∈N ∈ Φ such that lim γ
bi = x
i→∞
66
H.CHEN
CHENAND
ANDJ.-P.
J.-P.LABB
LABB
H.
ÉÉ
Dyeralso
alsostudied
studiedlimit
limitroots
rootsas
asthe
theboundary
boundaryofofthe
theimaginary
imaginarycone,
cone,see
see[Dye13,
[Dye13,Theorem
Theorem5.4].
5.4].
Dyer
Limit roots
roots are
are on
on the
the isotropic
isotropic cone
cone {b
{�
PV | | B(x,
B(x,x)
x) == 0},
0}, see
see [HLR14,
[HLR14, Theorem
Theorem 2.7(ii)].
2.7(ii)].
Limit
xx ∈∈ PV
Limitsroots
rootsare
arealso
alsolimit
limitdirections
directionsarising
arisingfrom
fromdifferent
differentbase
basepoints.
points. The
Thefollowing
followingtheorem,
theorem,
Limits
schematizedininFigure
Figure3,3,summarizes
summarizesthe
theknown
knownresults.
results.
schematized
Theorem 2.3.
2.3. Limit
Limit roots
roots ofof aa geometric
geometric Coxeter
Coxeter system
system (W,
(W,S)
S)BB are
are limit
limit directions
directions arising
arising
Theorem
from
from
(i)
(i) simple
simpleroots,
roots,see
see[HLR14,
[HLR14,Definition
Definition2.12],
2.12],
(ii)
(ii) limit
limitroots,
roots,see
see[DHR13,
[DHR13,Theorem
Theorem3.1(b)],
3.1(b)],
(iii)
(iii) projective
projectiveroots,
roots,see
see[DHR13,
[DHR13,Theorem
Theorem3.1(c)].
3.1(c)].
Moreover,
Moreover,ifif(W,
(W,S)
S)BB isisLorentzian,
Lorentzian,limit
limitroots
rootsare
arelimit
limitdirections
directionsarising
arisingfrom
from
(iv)
(iv) time-like
time-likedirections,
directions,see
see[HPR13,
[HPR13,Theorem
Theorem3.3],
3.3],
(v)
projective
weights,
see
[CL14,
Theorem
(v) projective weights, see [CL14, Theorem3.4],
3.4],
(vi)
(vi) light-like
light-likedirections,
directions,by
byTheorem
Theorem2.5
2.5ofofthis
thispaper.
paper.
R
P
W
S
L
T
Figure 3. Schematic picture showing a limit root P arising from different base
Figure
Schematic
picture (T),
showing
a limit
root P (S),
arising
from different
points:3.time-like
direction
light-like
direction
projective
weight base
(W),
points:
time-like
direction
(T),
light-like
direction
(S),
projective
weight
(W),
projective root (R) or another limit root (L). The elliptic-shape is the projective
projective
another
root (L).
elliptic-shape
is the projective
light cone,root
and(R)
the or
triangle
is limit
the convex
hullThe
of the
projective simple
roots.
light cone, and the triangle is the convex hull of the projective simple roots.
Remark 2.4.
2.4. In
Inthis
thispaper,
paper,we
werequire
requirethe
thesimple
simpleroots
rootsto
tobe
beaabasis
basisfor
foraabased
basedroot
rootsystem.
system.
Remark
However,
except
for
the
concept
of
weights,
all
the
arguments
and
results
of
this
paper
work
However, except for the concept of weights, all the arguments and results of this paper work inin
the more
more general
general setting
setting ofof [HLR14,
[HLR14, Dye13,
Dye13, DHR13],
DHR13], where
where the
the simple
simple roots
roots only
only needs
needs to
to be
be
the
positively
independent
but
not
necessarily
linearly
independent.
positively independent but not necessarily linearly independent.
2.3. Light-like
Light-likelimit
limitdirections.
directions. By
ByTheorem
Theorem2.3,
2.3,the
theset
setofoflimit
limitroots
rootsEEΦisiscontained
containedininEEV. .
2.3.
Φ
V
The
following
theorem
states
that
limit
roots
are
the
only
light-like
limit
directions
Lorentzian
The following theorem states that limit roots are the only light-like limit directions ofofLorentzian
Coxetergroups.
groups.
Coxeter
Theorem 2.5.
2.5. For
For aa Lorentzian
Lorentzian Coxeter
Coxeter system
system (W,
(W,S)
S)BB, , consider
consider aa sequence
sequence (w
(wkk))k∈N
W
k∈N ∈∈ W
Theorem
�
�
and
a
base
point
x
∈
V
.
If
(w
·
x
)
is
injective
and
converges
to
a
limit
direction
x
in
the
k∈N is injective and converges to a limit direction x
b00)k∈N
b in the
and a base point x00∈ V . If (wkk· x
� then x
projectivelight
lightcone
coneQ,
limitroot.
root.
bQ,
b�isisaalimit
projective
then x
Proof. Choose
Choose aa sequence
sequence ofof simple
simple roots
roots (α
(αkk))k∈N
such that
that (w
(wkk· ·ααkk))k∈N
also an
an injective
injective
k∈N such
k∈N isis also
Proof.
sequence
of
projective
positive
roots.
Since
∆
is
a
finite
set,
the
sequence
(α
)
visits
a
certain
k∈Nvisits a certain
sequence of projective positive roots. Since ∆ is a finite set, the sequence (αkk)k∈N
simple
root,
say
α
∈
∆,
infinitely
many
times.
By
passing
to
a
subsequence,
we
may
assume
that
simple root, say α ∈ ∆, infinitely many times. By passing to a subsequence, we may assume that
α
=
α
for
all
k
∈
N.
Since
PV
is
compact,
by
passing
again
to
a
subsequence,
we
may
assume
αkk = α for all k ∈ N. Since PV is compact, by passing again to a subsequence, we may assume
LIMIT DIRECTIONS FOR LORENTZIAN COXETER SYSTEMS
7
b Assume that x0 is not light-like, then h(wk (x0 ))
that (wk · α) converges to a limit root βb ∈ Q.
tends to infinity since
LIMIT DIRECTIONS FOR LORENTZIAN COXETER SYSTEMS
B(x0 , x0 )
b) = lim B(wk · x
b0 , wk · x
b0 ) = lim
.
0 = B(b
x, x
2
� Assume thatk→∞
k (x
0 ))
converges to a limit k→∞
root β� ∈ Q.
x0 ish(w
not
light-like,
7
that (wk · α)
then h(wk (x0 ))
tends to infinity since
The height h(wk (α)) tends to infinity (see the proof of [HLR14, Theorem 2.7]). Moreover,
B(x0 , x0 )
B(wk (α), wk (x0 )) =0B(α,
xx0 ,) x
constant.
Therefore
�is) =
� 0 , wk · x
�0 ) = lim
.
= B(�
lim B(w
k ·x
k→∞
k→∞ h(wk (x0 ))2
B(x0 , α)
b tends
The height h(wB(b
infinity
the=proof
2.7]). Moreover,
k (α))
b0 ,(see
x, β)
= limtoB(w
wk · α)
lim of [HLR14, Theorem
= 0.
k ·x
k→∞
k→∞ h(wk (x0 ))h(wk (α))
B(wk (α), wk (x0 )) = B(α, x0 ) is constant. Therefore
α) Proposition 2.1. The limit
b and βb are
bB(x
Since x
both
in the
cone,
we have x
= βb0 ,by
� =
�0 ,light
B(�
x
, β)
lim projective
B(wk · x
wk · α)
= lim
= 0.
k→∞
k→∞
h(w
(x
))h(w
k argument
0
k (α))
b is therefore a limit root. Furthermore, the above
direction x
does not depend on the
b is another limit direction
b∈Q
choice
base
x0 ∈
/ Q.
if y
arising
from y02.1.
∈
/ QThe
through
�ofand
� = β� by
Since x
β� point
are both
in
theSo
projective
light cone, we have x
Proposition
limit
b
b
b
the
same
sequence
(w
)
,
we
have
x
=
y
=
β
∈
E
.
k k∈N
Φ
� is therefore
direction x
a limit root. Furthermore, the
above argument does not depend on the
0
If x0ofisbase
light-like,
be So
decomposed
a linearlimit
combination
a time-like
vector
� isasanother
0 and a
�∈Q
choice
point it
x0can
∈
/ Q.
if y
directionofarising
from y
/ Qxthrough
0 ∈
00
space-like
vector
x
,
see
Figure
4
for
an
illustration
of
this
case.
Under
the
action
of
the
sequence
�=
� = β� ∈ EΦ .
the same sequence0(wk )k∈N , we have x
y
(wkIf)k∈N
time-likeitcomponent
(wk ·b
x00 )k∈N
to a limit root,
[HPR13,
Theorem
3.3].a
x0 ,isthe
light-like,
can be decomposed
as converges
a linear combination
of asee
time-like
vector
x�0 and
00
b
��
Let
β be this
limit
If the 4space-like
component
doesthe
notaction
converge
to sequence
the light
k (x
space-like
vector
x0 root.
, see Figure
for an illustration
of (w
this
case.
Under
of the
0 ))k∈N
00
cone,
the, the
norm
of wk (xcomponent
the actionto
ofaW
preserves
the[HPR13,
bilinearTheorem
form. In 3.3].
this
(wk )k∈N
time-like
(wk because
·�
x�0 )k∈N converges
limit
root, see
0 ) is bounded
case,
we
have
��
�
Let β be this limit root. If the space-like component (w (x ))
does not converge to the light
k
0
k∈N
cone, the norm of wk (x��0 ) is bounded
the w
action
the bilinear form. In this
b0 = lim
b00 of
lim wk because
·x
=W
βb ∈preserves
EΦ .
k ·x
k→∞
k→∞
case, we have
�0to=the
��0 =itsβ� direction
wk · x
lim
wk cone,
·x
∈ EΦ . (wk · x
b000 )k∈N also converges
If the space-like component lim
converges
light
k→∞
k→∞
b Then the sequence (wk · x
b0 )k∈N
to If
the
root β.
, being
the direction
�� a light-like linear
�of
thelimit
space-like
component converges to the light
cone,
its direction
(wk · x
0 )k∈N also converges
b
combination
of
the
two
components,
must
converge
to
the
same
limit
root
β.
�
� )
to the limit root β. Then the sequence (w · x
, being the direction of a light-like linear
k
0 k∈N
�
combination of the two components, must converge to the same limit root β.
As a consequence, limit directions arising from light-like directions are limit roots.
As a consequence, limit directions arising from light-like directions are limit roots.
���0
wk · x
�0
wk · x
���0
x
�0
x
��0
wk · x
�
β�
��0
x
Figure 4. Illustration for the proof of Theorem 2.5 in the case where the base
Figure 4. Illustration for the proof of Theorem 2.5 in the case where the base
point x0 is light-like.
point x0 is light-like.
Corollary 2.6 (of the proof). Limit roots arising from different base points but through the same
sequence are2.6
the(of
same.
Corollary
the proof). Limit roots arising from different base points but through the same
sequence
theIfsame.
Remark are
2.7.
the geometric Coxeter system (W, S)B is not Lorentzian, limit roots are in
general not the only limit directions on the isotropic cone. An example is given in Example 3.11
Remark
of Section 2.7.
3.3. If the geometric Coxeter system (W, S)B is not Lorentzian, limit roots are in
general not the only limit directions on the isotropic cone. An example is given in Example 3.11
2.4.Section
Infinite
of
3.3.reduced words. An infinite reduced word is an infinite sequence w = s1 s2 . . . of
generators in S such that every finite prefix wk = s1 s2 . . . sk is a reduced expression, see [LP13,
Section 4.2] and [LT13]. The inversion set inv(w) ⊂ Φ+ is the set of positive roots in the form
of wk−1 (αsk ), k ∈ N. The same definition also applies to finite words. To prove the following
theorem, we need the notion of biclosed sets of roots. A subset A of roots in Φ+ is closed if for
8
H. CHEN AND J.-P. LABBÉ
2.4. Infinite reduced words. An infinite reduced word is an infinite sequence w = s1 s2 . . . of
generators in S such that every finite prefix wk = s1 s2 . . . sk is a reduced expression, see [LP13,
Section 4.2] and [LT13]. The inversion set inv(w) ⊂ Φ+ is the set of positive roots in the form
of wk−1 (αsk ), k ∈ N. The same definition also applies to finite words. To prove the following
theorem, we need the notion of biclosed sets of roots. A subset A of roots in Φ+ is closed if for
two roots α, β ∈ A, any root that is a positive combination of α and β is also in A. A subset A is
biclosed if both A and its complement in Φ+ are closed. Finite biclosed sets of Φ+ are in bijections
with the inversion sets of elements of W , see [Pil06, Proposition 1.2]. See also [Lab13, Chapter 2]
for more detail on the relation between biclosed sets and the study of limit roots.
Theorem 2.8. Let w be an infinite reduced word of a Lorentzian Coxeter system (W, S)B . The
\ of an infinite reduced word w has a unique limit root as its accuprojective inversion set inv(w)
mulation point.
\ are limit roots. For the sake of
Proof. As a set of positive root, the accumulation points of inv(w)
\
b and y
b . Because the
contradiction, assume that inv(w) accumulates at two distinct limit roots x
\
b
projective light cone is strictly convex, we can find two projective roots α
b, β ∈ inv(w) respectively
b intersect Q.
b Moreover, there is a
b and y
b such that the segment [b
in the neighborhood of x
α, β]
positive integer k > 0 such that α and β are contained in inv(wk ). However, the reflections in
α and β generate an infinite dihedral group, so inv(wk ) can not be finite and closed at the same
time.
Thus, we can associate a unique limit root to an infinite reduced word w and denote it by γ
b(w).
Corollary 2.9. The limit root γ
b(w) arises through the sequence of prefixes (wk = s1 . . . sk )k∈N
of w.
Proof. At least one generator s ∈ S appears in w infinitely many times, so we can take from
inv(w) an injective subsequence (wk (αs )) such that sk+1 = s. Then γ
b(w) is the limit root arising
from αs through the sequence (wk ). By Corollary 2.6, the same limit root arises through the
same sequence from any projective root. We then conclude that γ
b(w) arises through the sequence
(wk )k∈N .
Consider two infinite reduced word w and w0 . It is easy to see that γ
b(w) = γ
b(w0 ) if inv(w) ∩
0
inv(w ) contains infinitely many roots. Lam and Thomas [LT13, Theorem 1(1)] proved that w
and w0 correspond to the same set of points on the Tits boundary of the Davis complex if inv(w)
and inv(w0 ) differ by finitely many roots. Moreover, since infinite reduced words correspond to
geodesic rays in the Cayley graph of (W, S), there is also a correspondence between limit roots
and the group ends of W . In Figure 5, the inversion set of an infinite Coxeter element and
the corresponding sequence of chambers in the Tits cone are shown. The sequence of chambers
correspond to a geodesic ray in the Cayley graph of (W, S).
It would be interesting to find an equivalence relation on infinite reduced words such that two
words w and w0 are equivalent if and only if γ
b(w) = γ
b(w0 ). None of the results above satisfy this
requirement. Consider the infinite reduced words (st)∞ and (ts)∞ of an affine infinite dihedral
group. Their inversion sets are disjoint, but they correspond to a same limit root. It would make
more sense if (st)∞ ∼ (ts)∞ whenever cst = 1. A way to encode geometric information (the
bilinear form B) into the equivalence relation would be very helpful. Besides, it may be possible
to define limit roots as the completion of limit roots obtained from infinite-order elements using a
metric on sequences of infinite-order elements which respects the geometry.
On the one hand, Lam and Pylyavskyy conjectured that the limit weak order, i.e. the finite
and infinite inversion sets ordered by inclusion, for the affine Coxeter group Ãn forms a lattice,
see [LP13, Conjecture 10.3]. On the other hand, Dyer conjectured that the extended weak order,
i.e. the biclosed sets ordered by inclusion, forms a complete ortholattice, see [Dye11, Conjecture 2.5]. In view of Theorem 2.8, it seems reasonable to use the notion of limit roots to unify
both conjectures. Namely, one verifies that an infinite inversion set inv(w) is biclosed in the affine
and Lorentzian case, otherwise it would contradict the biclosedness of the inversion set of a certain
LIMIT DIRECTIONS FOR LORENTZIAN COXETER SYSTEMS
LIMIT DIRECTIONS FOR LORENTZIAN COXETER SYSTEMS
9
9
γ
u
4
s
4
4
t
α
β
∞
Figure
Figure 5.
5. The
The inversion
inversion set
set of
of the
the infinite
infinite reduced
reduced word
word (stu)
(stu)∞ represented
represented by
by
the
convex
hull
of
its
roots,
and
the
corresponding
sequence
of
chambers
the convex hull of its roots, and the corresponding sequence of chambers inside
inside
the
the Tits
Tits cone.
cone. They
They are
are disjoint
disjoint and
and share
share aa unique
unique point
point on
on their
their boundary:
boundary: the
the
limit
root
given
by
the
dominant
eigendirection
of
stu.
limit root given by the dominant eigendirection of stu.
3. Spectra of elements of Lorentzian Coxeter groups
finite prefix of w. The difference between the two conjectures lies in the fact that there are many
3.1.
Spectra
of Lorentz
transformations.
In yet
a Lorentz
(V,inversion
B), a subspace
U [Lab13,
of V is
biclosed
sets that
are neither
finite nor cofinite,
are notspace
infinite
sets, see
space-like
if
its
non-zero
vectors
are
all
space-like,
light-like
if
U
contains
some
non-zero
Figure 2.11] for an example. The relations between the two conjectures should be made clearlightand
like
vector
but attention.
no time-like vector, or time-like if U contains some time-like vector. Two vectors
deserve
better
x, y ∈ V are said to be orthogonal if B(x, y) = 0. For a vector x ∈ V , we define its orthogonal
hyperplane
3. Spectra of elements
H = {y ∈of
V Lorentzian
| B(x, y) = 0}.Coxeter groups
x
3.1.see
Spectra
In a Lorentz
space (V,(resp.
B), alight-like,
subspacespace-like).
U of V is
We
that Hxofis Lorentz
space-like transformations.
(resp. light-like, time-like)
if x is time-like
space-like
if its Unon-zero
are all
space-like,
light-likeasif U contains some non-zero lightFor
a subspace
of V , itsvectors
orthogonal
companion
is defined
like vector but no time-like vector,
or time-like if U contains some time-like vector. Two vectors
U ⊥ = {y ∈ V | B(y, x) = 0 for all x ∈ U }.
x, y ∈ V are said to be orthogonal if B(x, y) = 0. For a vector x ∈ V , we define its orthogonal
If
U is lightlike, U + U ⊥ is not the whole space V .
hyperplane
To study the eigenvalues and eigenvectors
it is useful to work in
Hx = {y ∈ Vof |Lorentz
B(x, y) transformations,
= 0}.
the complexification VC = V ⊕ iV . A vector z ∈ VC can be written as z = x + iy for x, y ∈ V . We
We see
that H
is space-like
if xand
is time-like
(resp.
space-like).
call
x (resp.
y)x the
real part (resp.
(resp.light-like,
imaginarytime-like)
part) of z,
the vector
z = light-like,
x − iy represents
its
For
a
subspace
U
of
V
,
its
orthogonal
companion
is
defined
as
conjugate vector. If y �= 0, we say that z is space-like (resp. light-like, time-like) if the subspace
spanned by x and y is space-like
(resp. light-like, time-like). The bilinear form B on V is viewed
U ⊥ = {y ∈ V | B(y, x) = 0 for all x ∈ U }.
as the restriction of a sesquilinear form on VC defined by requiring in addition that
If U is lightlike, U + U ⊥ is not the B(λz
whole ,space
V.
1 µz2 ) = λµB(z1 , z2 )
To study the eigenvalues and eigenvectors of Lorentz transformations, it is useful to work in
for
the action
the Lorentz group O x(V
1 , z2 ∈ VC and λ,
the zcomplexification
VCµ=∈V C.
⊕ iVThen
. A vector
z ∈ Vof
+ )iynaturally
for x, y ∈extends
V . We
C can be written as z = B
to
V
.
Again,
two
vectors
z
,
z
∈
V
are
orthogonal
if
B(z
,
z
)
=
0.
2
C
1 the
2
call Cx (resp. y) the real part1 (resp.
imaginary
part) of z, and
vector z = x − iy represents its
conjugate 3.1.
vector.
y 6= 0,form
we say
that z is to
space-like
(resp.
light-like,
time-like)
if the
Remark
TheIfbilinear
B associated
V can also
be viewed
as the
restriction
of asubspace
bilinear
spanned
by
x
and
y
is
space-like
(resp.
light-like,
time-like).
The
bilinear
form
B
on
V is viewed
form on VC , as in [Rie58, Chapter III]. This is algebraically more natural, while a sesquilinear
form
asgeometrically
the restrictionmore
of a natural.
sesquilinear form on VC defined by requiring in addition that
is
B(λz1 , µz2of) =
λµB(z1 ,transformation
z2 )
A non-zero vector z ∈ VC is an eigenvector
a Lorentz
φ ∈ OB (V ) if φ(z) = λz
for
some
λ
∈
C.
An
eigenvalue
λ
is
unimodular
if
|λ|
=
1,
in
which
case
a
is also
for z1 , z2 ∈ VC and λ, µ ∈ C. Then the action of the Lorentz group OBλ-eigenvector
(V ) naturally zextends
�
said
to
be
unimodular.
If
x
∈
V
is
an
eigenvector
of
φ,
we
call
x
∈
PV
an
eigendirection
of
φ.
to VC . Again, two vectors z1 , z2 ∈ VC are orthogonal if B(z1 , z2 ) = 0.
The following proposition gathers some basic facts about eigenvectors.
10
H. CHEN AND J.-P. LABBÉ
Remark 3.1. The bilinear form B associated to V can also be viewed as the restriction of a bilinear
form on VC , as in [Rie58, Chapter III]. This is algebraically more natural, while a sesquilinear form
is geometrically more natural.
A non-zero vector z ∈ VC is an eigenvector of a Lorentz transformation φ ∈ OB (V ) if φ(z) = λz
for some λ ∈ C. An eigenvalue λ is unimodular if |λ| = 1, in which case a λ-eigenvector z is also
b ∈ PV an eigendirection of φ.
said to be unimodular. If x ∈ V is an eigenvector of φ, we call x
The following proposition gathers some basic facts about eigenvectors.
Proposition 3.2. Let φ be a Lorentz transformation and z be a λ-eigenvector of φ, then
(i) z is an eigenvector of φ with eigenvalue λ,
(ii) z is an eigenvector of wk , k ∈ N, with eigenvalue λk ,
(iii) z is an eigenvector of w−1 with eigenvalue λ−1 ,
(iv) let ϕ ∈ OB (V ), then ϕ(z) is an eigenvector of ϕφϕ−1 with eigenvalue λ.
Proposition 3.3. Let z1 and z2 be λ- and µ-eigenvectors of φ ∈ OB (V ), respectively. If λµ 6= 1,
then B(z1 , z2 ) = 0.
Proof. Since w preserves the bilinear form, we have
B(z1 , z2 ) = B(φ(z1 ), φ(z2 )) = λµB(z1 , z2 ).
So B(z1 , z2 ) = 0 because λµ 6= 1.
In the following propositions, we classify Lorentz transformations into three types. Such a
classification is present in many references, often in the language of Möbius transformations or
hyperbolic isometries, see for instance [AVS93, Chapter 4, Theorem 1.6], [Rat06, Section 4.7],
[Kra09, Proposition 4.5.1] and [SR13, Section 7.8]. Our formulation is adapted from [Rie58,
Chapter III], which deals with Lorentz space and is suitable for our use. See also discussions
in [Cec08, Section 3.3] for a geometric insight.
Proposition 3.4 ([Rie58, Section 3.7]). Lorentz transformations are partitionned into three types:
• Elliptic transformations are diagonalizable, and have only unimodular eigenvalues,
• Parabolic transformations have only unimodular eigenvalues, but are not diagonalizable,
• Hyperbolic transformations are diagonalizable and have exactly one pair of simple, real,
non-unimodular eigenvalues, namely λ±1 for some λ > 1.
Proposition 3.5 ([Rie58, Section 3.7-3.9]). The two non-unimodular eigendirections of a hyperbolic transformation are light-like, while its unimodular eigenvectors are space-like.
Proposition 3.6 ([Rie58, Section 3.10]). The Jordan form of a parabolic transformation φ contains a unique Jordan block of size 3, corresponding to the eigenvalue ε = 1 or −1. The (n − 2)dimensional real subspace Uφ spanned by eigenvectors of φ is light-like. The 1-dimensional lightlike subspace of Uφ is a ε-eigendirection. The minimal polynomial f (x) such that f (φ) annihilates Uφ⊥ is (x − ε)2 .
3.2. Spectral interpretation of limit roots. Let (W, S)B be a Lorentzian Coxeter system.
In this part, we consider the limit directions arising through sequences in the form of (wk )k∈N
for some w ∈ W . We say that w is an elliptic (resp. parabolic, hyperbolic) element of W if its
corresponding transformation ρ(w) is an elliptic (resp. parabolic, hyperbolic) transformation.
Theorem 3.7. An element w of a Lorentzian Coxeter system (W, S)B is of finite order if and
only if w is an elliptic Lorentz transformation.
Proof. Assume that wk = e for some k < ∞. Since the minimal polynomial of w divides xk − 1, its
roots are all distinct and unimodular. Hence w is diagonalizable with only unimodular eigenvalues,
i.e. w is elliptic.
For the sake of contradiction, assume that w is an elliptic element but of infinite order. Let
the sequence (wk )k∈N act on a simple root α ∈ ∆. Since w is diagonalizable and every eigenvalue
is unimodular, we conclude that α
b is itself a limit direction. However, by [HLR14, Theorem 2.7],
limit directions arising from simple roots are on the light cone and therefore can not be a root. LIMIT DIRECTIONS FOR LORENTZIAN COXETER SYSTEMS
11
Consequently, whenever w is of infinite order, it is either parabolic or hyperbolic. Then Theorem 1.1 follows directly from the fact that the set of limit roots equals to the limit set of the
Coxeter group regarded as a Kleinian group acting on the hyperbolic space [HPR13, Theorem 1.1].
For the relation between fixed points and limit sets of Kleinian groups, see for instance [Mar07,
Lemma 2.4.1(ii)].
However, we provide here a different proof for the following reasons. Firstly, in the proof of
Theorem 3.8 and 3.9, the behavior of infinite-order elements is analysed in detail. This will be
useful in the proof of Theorem 1.2. Secondly, our proof is independent of [HPR13, Theorem 1.1].
In fact, as shown by Theorem 1.2 and Example 3.11, limit directions arising from a space-like
and a time-like base point are not necessarily the same. So the equality between the set of limit
roots and the limit set of Coxeter group is not trivial. As mentioned in the introduction, our
proof tries to rely minimally on hyperbolic geometry, in the hope of a deeper insight and further
generalizations to non-Lorentzian infinite Coxeter systems. In particular, concepts involved in the
statement of Theorem 1.1 are all well defined for general infinite Coxeter systems. The possibility
of a generalization is discussed in detail in Section 3.3.
For a Lorentzian Coxeter group W , denote by W∞ the set of elements of infinite order, Wpar
the set of parabolic elements, and Whyp the set of hyperbolic elements. Then
W∞ = Wpar t Whyp .
Given an element w ∈ W∞ , the (n − 2)-dimensional real subspace spanned by unimodular eigenvectors of w is called the unimodular subspace of w, and denoted by Uw .
b of a parabolic element w ∈ Wpar is a limit root
Theorem 3.8. The light-like eigendirection x
of W .
12
H. CHEN AND J.-P. LABBÉ
�
Q
�
e2
�
e1
�
x
�w⊥
U
Figure 6. The dynamics of a parabolic element as described in the proof of TheFigure 6. The dynamics of a parabolic element as described in the proof of Theorem 3.8.
orem 3.8.
w is diagonalizable, there exists an eigenbasis of VC consisting of x, x− and n − 2 unimodular
Proof.
Let x be
the Then
light-like
ε = 1into
or −1. By Proposition 3.6,
eigenvectors
of w.
any eigenvector
real vector yof∈wV with
can eigenvalue
be decomposed
there are two vectors e1 , e2 ∈
/ Uw such that w(e2 ) − εe2 = e1 and w(e1 ) − εe1 = x. Note that e1
−
a+ xvector,
+ a− xit
+
y◦be
, decomposed into
is(5)
in Uw⊥ , while e2 is not. Let y ∈ V bey a=real
can
−
◦3.2(ii), we have
where y◦ ∈ Uw is orthogonal to x and
(3)
y =xax. +By
beProposition
1 + ce2 + y
k
k
−k −
◦k
a+ λx.
xUnder
+ a− λthe
xaction
+ wkof
(yw
) , we have
where y◦ ∈ Uw is orthogonal towe1(y)
, e2=and
Using y as the base point, three
cases
possible
(4)
wk (y)
= aare
bk e1 + ck e2 + wk (y◦ ).
kx +
Case 1: a+ �= 0, so y ∈
/ Hx− . The coefficient a+ λk diverges to infinity while the coeffi− −k
cient a λ
tends to 0. Besides, the term wk (y◦ ) has a bounded norm since y◦ ∈ Uw .
Therefore, as k tends to infinity, the direction of the sequence (wk (y))k∈N converges to
� is a limit direction arising from y
� through the sequence (wk )k∈N .
the direction of x. So x
� �= x
�− . Again, the coefficient a− λ−k tends
Case 2: a+ = 0 and y◦ �= 0, so y ∈ Hx− but y
to 0, so y◦ ∈ Uw is an accumulation point of the set {wk (y) | k ∈ N}. Consequently,
� ◦ ∈ Uw is a limit direction arising from y
� through some subsequence of (wk )k∈N .
y
�=x
�− . The sequence of vectors (wk (y)) converges
Case 3: a+ = 0, y◦ = 0 and a− �= 0, so y
�
e1
12
�w⊥
U
�
x
H. CHEN AND J.-P. LABBÉ
where the coefficients
k−2
k−1
Figure 6. The dynamics aof =
a parabolic
in the proof of Theaεk + bkεelement
+ c k2asεdescribed
,
k
orem 3.8.
bk = bεk + ckεk−1 ,
k
. coefficient a+ λk diverges to infinity while the coeffiCase 1: a+ �= 0, so y ∈
/ Hcxk−=. cε
The
−
−k
◦
a ◦ )λ hastends
to 0.norm
Besides,
wk (y◦ ) in
hasthe
a bounded
norm
since yU
The termcient
wk (y
bounded
sincethe
y◦ term
is contained
unimodular
subspace
As
w.
w∈. U
k
k tends
thethen
direction
of the
sequence
(w
(y))
converges
to
long as yTherefore,
6= Uw , weashave
b 6= to0 infinity,
or c 6= 0,
bk = o(a
)
and
c
=
o(b
),
i.e.
a
dominates
k∈N
k
k
k
k
k
�
�
direction
of
x.
So
x
is
a
limit
direction
arising
from
y
through
the
sequence
(w
)
.
bk and cthe
,
and
b
dominates
c
,
as
k
tends
to
infinity.
Consequently,
the
direction
of
the
sek∈N
k
k
k
+
� �= x
�is,− .x
− but
2: ak∈N
=converges
0 and y◦to�= the
0, so
y ∈ Hxof
y
thedirection
coefficient
a− λ−kfrom
tends
b Again,
b
quenceCase
(wk (y))
direction
x. That
is a limit
arising
y
so y◦ ∈(w
Ukw)k∈N
is an
through to
the0,sequence
. accumulation point of the set {wk (y) | k ∈ N}. Consequently,
◦
k
�
� through
∈ Uw the
is a vector
limit direction
y
)k∈N
.
Since y
∆ spans
space V , arising
there isfrom
a simple
root αsome
∈
/ Uw subsequence
. Using α as of
the(w
base
point,
we
+
◦
−
−
k
�=
� . The sequence
Case
3:x
0, y root.
= 0 and
�= 0, so of
y
x
vectors (win(y))
converges
bais a=limit
conclude
that
Theadynamics
a parabolic
element isofillustrated
Figure
6. � remains x
�− for all k ∈ N. The sequence (wk · x
�) visits
to 0, while the direction of wk · y
Theorem
3.9.
Let
w
∈
W
be
a
hyperbolic
element.
The two light-like eigendirections of w are
hyp
only one point in PV , so no limit direction arises.
bw is contained in the set of limit directions
limit roots and the projective unimodular subspace U
Since ∆ spans the vector space V , there is a simple root
α∈
/ Hx− . Using α as the base point, we
of W .
� is a limit root.
conclude from Case 1 that x
�
�−
x
�
Q
�w
U
�
x
Figure 7. The dynamics of a hyperbolic element as described in the proof of
Figure
7. 3.9.
The dynamics of a hyperbolic element as described in the proof of
Theorem
Theorem 3.9.
The dynamics of a hyperbolic element is illustrated in Figure 7.
Proof.
Proposition
3.4,system
the element
possesses
a light-like
x which eigendiis nonFor From
a Lorentzian
Coxeter
(W, S)Bw, let
Epar (resp.
Ehyp ) beeigenvector
the set of light-like
−1
unimodular.
By
replacing
w
with
w
if
necessary,
we
may
assume
that
the
eigenvalue
λ
correrections of elements in Wpar (resp. Whyp ). We now prove the following theorem, which is equivalent
sponding to x is greater than 1. Let x− be an eigenvector of w with eigenvalue λ−1 . Since w is
diagonalizable, there exists an eigenbasis of VC consisting of x, x− and n − 2 unimodular eigenvectors of w. Then any real vector y ∈ V can be decomposed into
(5)
y = a+ x + a− x− + y◦ ,
where y◦ ∈ Uw is orthogonal to x and x− . By Proposition 3.2(ii), we have
wk (y) = a+ λk x + a− λ−k x− + wk (y◦ )
Using y as the base point, three cases are possible
Case 1: a+ 6= 0, so y ∈
/ Hx− . The coefficient a+ λk diverges to infinity while the coeffi− −k
cient a λ
tends to 0. Besides, the term wk (y◦ ) has a bounded norm since y◦ ∈ Uw .
Therefore, as k tends to infinity, the direction of the sequence (wk (y))k∈N converges to
b is a limit direction arising from y
b through the sequence (wk )k∈N .
the direction of x. So x
+
◦
−
b 6= x
b . Again, the coefficient a− λ−k tends
Case 2: a = 0 and y 6= 0, so y ∈ Hx− but y
◦
to 0, so y ∈ Uw is an accumulation point of the set {wk (y) | k ∈ N}. Consequently,
b ◦ ∈ Uw is a limit direction arising from y
b through some subsequence of (wk )k∈N .
y
LIMIT DIRECTIONS FOR LORENTZIAN COXETER SYSTEMS
13
b=x
b− . The sequence of vectors (wk (y)) converges
Case 3: a+ = 0, y◦ = 0 and a− 6= 0, so y
b remains x
b− for all k ∈ N. The sequence (wk · x
b) visits
to 0, while the direction of wk · y
only one point in PV , so no limit direction arises.
Since ∆ spans the vector space V , there is a simple root α ∈
/ Hx− . Using α as the base point, we
b is a limit root. The dynamics of a hyperbolic element is illustrated
conclude from Case 1 that x
in Figure 7.
For a Lorentzian Coxeter system (W, S)B , let Epar (resp. Ehyp ) be the set of light-like eigendirections of elements in Wpar (resp. Whyp ). We now prove the following theorem, which is equivalent
LIMIT
DIRECTIONS
FOR LORENTZIAN
COXETER
SYSTEMS
13
DIRECTIONS
FOR LORENTZIAN
COXETER
SYSTEMS
13
to Theorem 1.1 since ELIMIT
∞ = Ehyp t Epar . It can be derived from the minimality of EΦ under the
action of W [DHR13, Theorem 3.1(b)]. We provide here a self-contained proof using the results
to Theorem
1.1 since
E∞
=E
� .EIt
. It be
canderived
be derived
the minimality
EΦ under
to Theorem
1.1 since
E∞ =
Ehyp
�hyp
Epar
from from
the minimality
of EΦofunder
the the
parcan
in Section
2.3.
action
W [DHR13,
Theorem
3.1(b)].
We provide
a self-contained
the results
action
of Wof[DHR13,
Theorem
3.1(b)].
We provide
here here
a self-contained
proofproof
usingusing
the results
in 3.10.
Section
2.3. a Lorentzian Coxeter system (W, S) , the set of light-like eigenvectors E
in Section
2.3. For
Theorem
B
par
(if not
empty)
and 3.10.
EFor
are
in
the
set
ofsystem
limit
roots
E,set
. ofset
Theorem
adense
Lorentzian
Coxeter
(W,
of light-like
eigenvectors
Theorem
3.10.
Lorentzian
Coxeter
system
(W, S)
the
light-like
eigenvectors
Epar Epar
B
B , S)
hypaFor
Φthe
(ifempty)
not empty)
and
are dense
theofset
of limit
(if not
and E
are
in theinset
limit
rootsroots
EΦ . EΦ .
hypdense
hyp E
Proof. We first prove that Ehyp is not empty. Since the group is infinite, the set of limit root EΦ
Proof.
We prove
first
that
not empty.
Since
the
group
is infinite,
theof
set
ofαlimit
EΦ that
We first
that
Ehyp E
ishyp
notissince
empty.
the V
group
is infinite,
the set
limit
root
EΦsuch
b beprove
is notProof.
empty.
Let
x
a limit
root,
∆Since
spans
, there
is
a simple
root
∈ ∆root
�a be
�Let
not empty.
a limit
∆ spans
V , there
is a simple
∆ such
is notis empty.
Let x
be x
limit
root,root,
sincesince
∆ spans
V , there
is a simple
root root
α ∈α
∆ ∈such
that that
by�different
b. two
B(α, x)
6=x)0,�=x)
and
the
reflection
in
α gives
gives
a limit
root
y
from
x
Take
two projective
�. Take
� different
�. Take
0, the
and reflection
the reflection
α gives
a limit
different
x
two projective
B(α, B(α,
0, �=and
in α in
a limit
root root
y
from from
x
projective
b
b
b
b, ,such
rootsroots
α
b and
β α�respectively
close
to
and
y
such
that
[b
intersect
light
�β]intersect
� [�
�y
� , such
� to
�and
roots
and
β� respectively
x
y
that
thesegment
segment
αα
, ,β]
thethe
light
conecone
α
� and
β�
respectively
close close
to x
x
and
that
thethe
segment
[�
α, β]
intersect
the light
cone
at two
the
of
reflections
α αand
ββ give
an hyperbolic
hyperbolic
element
at points.
two Then
points.
Then
the product
the reflections
givehyperbolic
an
element
at points.
two
Then
theproduct
product
of the
theofreflections
in in
α in
and
βand
give
an
element
in Win
. Win. W .
γ
�Φ∈be
aroot
limit
root
obtained
an
injective
sequence
(�
γγiiof
))i∈N
ofofprojective
By By
γ
�Let
∈
a be
limit
rootobtained
obtained
from
ananinjective
sequence
(�
γi )i∈N
projective
roots.roots.
By
Let γ
bLet
∈E
be
aElimit
fromfrom
injective
sequence
(b
projective
roots.
Φ
ΦE
i∈N
to a subsequence,
we assume
may
assume
γ
� obtained
is obtained
from
an injective
sequence
(gii∈N
(α))
passing
a subsequence,
may
assume
that
γ
�γ
obtained
fromfrom
an injective
sequence
(gi (α))
i∈N , i∈N ,
passing
topassing
a to
subsequence,
wewemay
thatthat
bis is
an
injective
sequence
(g,i (α))
where
is simple
a fixed
simple
root
∆ gand
giW
αa isfixed
aα fixed
simple
rootin
in ∆
∆ in
and
.∈ .W .
wherewhere
α is
root
and
gi i∈∈W
�
z hyp
∈ Ebe
a light-like
eigendirection
a hyperbolic
element
∈ W
By ProposiLet �
zLet
∈ E
light-like
eigendirection
of a of
hyperbolic
element
w ∈w
Whyp
. hyp
By. Proposihypa be
Let b
z ∈ Ehyp be� a light-like
eigendirection of a hyperbolic element −1
w ∈−1W
hyp . By Proposition 3.2(iv),
zlight-like
is a light-like
eigendirection
the hyperbolic
element
.
So
the
sequence
tion 3.2(iv),
gk · zgis
eigendirection
of theofhyperbolic
element
gk wggkk wg
. So
the
sequence
k ·a�
k −1
b
tion 3.2(iv),
·k∈N
zaissequence
eigendirection
the
hyperbolic
element
gk wg
. So
�
�
·g�
zk)is
isaalight-like
sequence
of
limit
in of
. compactness,
By
compactness,
we may
assume
that
·�
z)
(gk · (g
z)kk∈N
of limit
rootsroots
in Ehyp
.Ehyp
By
we
may
assume
(gkthe
· (g
z)ksequence
kthat
(gk · b
zconverges.
)k∈N
is aBysequence
of
roots
. root.
By compactness, we may assume that (gk · b
z)
converges.
By Theorem
2.5,limit
its
limit
is E
a hyp
limit
Theorem
2.5,limit
its
is a in
limit
root.
�
TheTheorem
last step
consists
oflimit
applying
tosequence
the sequence
·�
z)to
to prove
� is
The By
last
step
consists
applying
Corollary
to2.6the
(gk · (g
z)kk∈N
prove
that that
γ
� is γ
converges.
2.5,of its
is a Corollary
limit2.6
root.
k∈N
thestep
limit.
For this,
remains
to Corollary
prove
the sequence
contains
infinitely
distinct
points.
The γ
thelast
limit.
Forconsists
this,
it remains
to prove
that that
the sequence
contains
infinitely
points.
Thethat
The
ofit applying
2.6
to the
sequence
(gk distinct
·b
z)k∈N
to prove
b is
Coxeter
graph
of
a
Lorentzian
Coxeter
system
consists
of
irreducible
finite
Coxeter
graphs
and The
Coxeter
graph
of
a
Lorentzian
Coxeter
system
consists
of
irreducible
finite
Coxeter
graphs
and
the limit. For this, it remains to prove that the sequence contains infinitely distinct points.
one irreducible
Coxeter
of Lorentzian
The geometric
representations
of finite
Coxeter
one irreducible
Coxeter
graphgraph
of Lorentzian
type.type.
The geometric
representations
of finite
Coxeter
Coxeter
graph
of a Lorentzian
Coxeter
system
consists
of irreducible
finite
Coxeter
graphs
and
and irreducible
Lorentzian
Coxeter
groups
act irreducibly
onspace,
the space,
see [Hum92,
Propogroupgroup
and irreducible
Lorentzian
Coxeter
groups
act irreducibly
on the
see [Hum92,
Propoone irreducible
Coxeter
graph
of
Lorentzian
type.
The
geometric
representations
of
finite
Coxeter
6.3] [Vin71,
and [Vin71,
Lemma
this irreducible
action,
we find
can afind
a basis
sitionsition
6.3] and
Lemma
14]. 14].
UsingUsing
this irreducible
action,
we can
basis
b1 , . .b. 1, ,b. n. . , bn
groupofand
Coxeter
groups
actsequence
irreducibly
on the
space,
seevector
[Hum92,
PropoVtheinorbit
the orbit
·�
z.the
Byinjectivity
the injectivity
the
gi , there
exists
a basis
bi , such
V of
inirreducible
WLorentzian
·�
zW
. By
of theofsequence
gi , there
exists
a basis
vector
bi , such
�injective.
�gikis
sitionthat
6.3]that
[Vin71,
Lemma
14].
Using
this
action,
can
find a basis b1 , . . . , bn
·b
Taking
theirreducible
vector
z above
finishes
the
proof.
gkand
·b
Taking
this bthis
vector
z above
finishes
thewe
proof.
i is injective.
i as
i asbthe
The
work,
mutatis
mutandis,
for parabolic
elements.
arguments
work,
mutatis
mutandis,
forsequence
parabolic
� bi�
b
of V in The
the same
orbitsame
W ·arguments
z. By
the
injectivity
of the
gelements.
exists a basis vector
, such
i , there
b
that gk Examples
· bi Examples
is injective.
Taking
this bi asofthe
z above
finishes
the
proof.
of light-like
eigendirections
ofvector
parabolic
and hyperbolic
elements
are illustrated
of light-like
eigendirections
parabolic
and
hyperbolic
elements
are illustrated
in in
Figure
Figure
8 for
8the
forrank
the rank
3work,
universal
3 universal
Coxeter
Coxeter
groupgroup
with
the
classical
the classical
geometric
geometric
representation.
representation.
Ob- Ob- The
same
arguments
mutatis
mutandis,
forwith
parabolic
elements.
number
and distribution
are quite
different.
serveserve
that that
their their
number
and distribution
are quite
different.
(a)12The
12 light-like
eigendirections
of parabolic(b) The
Thelight-like
126 light-like
eigendirections
of hyperThe
light-like
eigendirections
of parabolic
126
eigendirections
of hyper(a)(a)
The
12 light-like
eigendirections
of parabolic
(b)(b)
The
126
light-like
eigendirections
of hyperelements
of length
smaller
or equal
bolic elements
of length
smaller
or equal
elements
of length
smaller
or equal
to 6. to 6.
bolic elements
of length
smaller
or equal
to 6. to 6.
elements of length smaller or equal to 6.
bolic elements of length smaller or equal to 6.
Figure
8. Light-like
eigendirections
of some
elements
the classic
universal
Figure
8. Light-like
eigendirections
of some
elements
of theofclassic
universal
geo- geo-
Figure
8.metric
Light-like
eigendirections
elements
of
the
classic
Coxeter
group
of rank
3,ofseen
in affine
the
affine
spanned
byuniversal
the simplegeometric
Coxeter
group
of rank
3, seen
insome
the
spacespace
spanned
by the
simple
metric
Coxeter
group of rank 3, seen in the affine space spanned by the simple
roots.
roots.
roots.
3.3. Non-Lorentzian
Coxeter
systems.
The density
of EinhypEΦincan
EΦ also
can be
alsoderived
be derived
3.3. Non-Lorentzian
Coxeter
systems.
The density
of Ehyp
from from
a result
of Conze
and Guivarc’h
[CG00].
In that
paper,
a linear
transformation
is said
a result
of Conze
and Guivarc’h
[CG00].
In that
paper,
a linear
transformation
is said
to beto be
proximal
if there
a simple
eigenvalue
is strictly
greater
any other
eigenvalue
proximal
if there
existsexists
a simple
eigenvalue
whichwhich
is strictly
greater
than than
any other
eigenvalue
in in
14
H. CHEN AND J.-P. LABBÉ
Examples of light-like eigendirections of parabolic and hyperbolic elements are illustrated in
Figure 8 for the rank 3 universal Coxeter group with the classical geometric representation. Observe that their number and distribution are quite different.
3.3. Non-Lorentzian Coxeter systems. The density of Ehyp in EΦ can also be derived from
a result of Conze and Guivarc’h [CG00]. In that paper, a linear transformation is said to be
proximal if there exists a simple eigenvalue which is strictly greater than any other eigenvalue in
absolute value. This is the case for hyperbolic transformations of Lorentzian Coxeter systems.
An eigenvector with this eigenvalue is called dominant eigenvector, so Ehyp is in fact the set of
dominant eigendirections. By [CG00, Proposition 2.4], Ehyp is the only minimal set of the action
of W . Then by [DHR13, Theorem 3.1(b)], this set is EΦ .
For a non-Lorentzian infinite Coxeter system (W, S), it remains true that the set of dominant
eigendirections is dense in the set of limit roots. It is proved in [Kra09, Section 6.5] that an element
w ∈ W is proximal if it is not contained in any proper parabolic subgroup of W .
However, an arbitrary element w of infinite order is not necessarily proximal, even if w has nonunimodular eigenvalues. Let λ denotes the eigenvalue of w with largest absolute value, it is possible
that the geometric multiplicity of λ is not 1. The real subspace U spanned by λ-eigenvectors of
b is
b in U
w is totally isotropic, meaning that B(x, y) = 0 for any x, y ∈ U . Then any direction x
an isotropic limit direction, but only those in conv(∆) are limit roots. Therefore, Theorem 1.1
and 2.5 do not generalize to other infinite Coxeter systems.
Example 3.11. In Figure 9, we show an example inspired by [HLR14, Example 5.8] and [Dye13,
Example 7.12 and 9.18]. It is the Coxeter graph of an irreducible Coxeter group, associated with
a non-degenerate bilinear form of signature (3, 2). We adopt Vinberg’s convention for Coxeter
graphs to encode both the Coxeter system (W, S) and the matrix B. The element s1√
s2 s4 s5 is
of infinite order with a simple eigenvalue 1, and two non-unimodular eigenvalues 7 ± 4 3, each
with multiplicity 2. Therefore, its decomposition into eigenspaces do not fall into any of the types
shown in Proposition 3.4.
s1
−2
s2
s3
s4
−2
s5
Figure 9. The Coxeter graph of a non-Lorentzian irreducible Coxeter group for
which the limit roots are not the only limit directions on the isotropic cone.
4. Coxeter arrangement and limit directions
In this section, we characterize the set of limit directions of a Lorentzian Coxeter system in
terms of the Coxeter arrangement in projective space.
4.1. Projective Lorentzian Coxeter arrangement. Let (W, S)B be a Lorentzian Coxeter system and (Φ, ∆) be the associated root system. The hyperplane Hγ orthogonal to a positive root
γ ∈ Φ+ is time-like and fixed by the reflection σγ . The projective Lorentzian Coxeter arrangement
is the set of projective hyperplanes
b γ | γ ∈ Φ+ }.
H = {H
Clearly, H is invariant under the action of W .
b ∈ PV to the sign
Let γ ∈ Φ+ be a positive root, the sign map sgnγ : PV → {+, −, 0} sends x
b ∈ PV is associated a sign sequence (sgnγ (b
of B(x, γ). To every direction x
x))γ∈Φ+ indexed by the
positive roots. The set of points with a fixed sign sequence is called a cell. The projective space PV
is therefore decomposed into cells. The set of cells is denoted by Σ. For two cells C, C 0 ∈ Σ, if
the sign sequence of C 0 is obtained from the sign sequence of C by changing zero or more (maybe
LIMIT DIRECTIONS FOR LORENTZIAN COXETER SYSTEMS
15
infinitely many) +’s or −’s to 0, we say that C 0 is a face of C, and write C 0 ≤ C. This defines a
partial order on Σ. The support of a cell C is defined as
\
bγ
supp(C) =
H
γ∈Φ+
sgnγ (C)=0
Cells are open in their supports. The dimension of a cell is defined as the dimension of its support.
The codimension of a cell is defined similarly. A cell is said to be space-like (resp. light-like, timelike) if its support is a projective space-like (resp. light-like, time-like) subspace. Cells with no 0
in their sign sequences are called chambers, they are connected components of the complement
b γ . Cells with exactly one 0 in their sign sequences are called panels, they are timePV \ ∪γ∈Φ+ H
like codimensional 1 cells. The projective Tits cone T is the union of cells whose sign sequences
contain finitely many −’s. In the Lorentzian case, the Tits cone is the convex cone over the set of
weights Ω, and contains the light cone [Max82, Corollary 1.3].
Remark 4.1. A Lorentzian hyperplane arrangement is infinite and not discrete. Unlike finite
hyperplane arrangements, the union of hyperplanes in a Lorentzian hyperplane arrangement is in
general not a closed set.
b intersect
4.2. Unimodular subspaces. Let α, β ∈ Φ be two positive roots. If the segment [b
α, β]
b
the projective light cone Q transversally (i.e. B(α, β) < −1), then σα σβ is a hyperbolic transb is tangent to Q
b (i.e. B(α, β) = −1), then σα σβ is a parabolic
formation. If the segment [b
α, β]
transformation. In either case, we know from [HLR14, Section 4] that the limit roots of the subb By Theorem 3.9, these are the light-like
b ∩ [b
group generated by σα and σβ are the points in Q
α, β].
eigendirections of σα σβ ∈ W∞ . The unimodular subspace of σα σβ is clearly Hα ∩ Hβ . We define
[
[
bα ∩ H
bβ ,
bα ∩ H
bβ ,
Lhyp =
H
Lpar =
H
α,β∈Φ+
B(α,β)<−1
α,β∈Φ+
B(α,β)=−1
Furthermore, we define the unions of projective unimodular subspaces for parabolic, hyperbolic,
and infinite-order elements.
[
[
[
bw
bw ,
bw ,
U
U
Uhyp =
U
U = Upar t Uhyp =
Upar =
w∈Wpar
w∈Whyp
w∈W∞
We have clearly Lhyp ⊂ Uhyp and Lpar ⊂ Upar . The following theorem concerns a reversed
inclusion.
bw of an element of infinite order w ∈ W∞
Theorem 4.2. The projective unimodular subspace U
is included in Lhyp . In other words,
U ⊆ Lhyp .
b to the
b) ∈ Q
Proof. Let Λ be the continuous map that sends a pair of light-like directions (b
x, y
bx ∩ H
by .
codimension-2 projective subspace H
b ) be its two non-unimodular eigendirections, then
For a hyperbolic element w ∈ Whyp , let (b
x, y
b
b ) = Uw . Since x
b and y
b are limit roots, we can find two sequences of roots (αk )k∈N and
Λ(b
x, y
bw and βbk converges to y
b . The sequence of segments [b
(βk )k∈N such that α
bk converges to x
αk , βbk ]
b at two limit roots, say x
bk and y
bk . These two limit
eventually intersect the projective light cone Q
b
bk ) ∈ Lhyp . The two sequences
roots determine a unimodular projective subspace Uk = Λ(b
xk , y
bk
b and y
b respectively. By the continuity of Λ, U
of limit roots (b
xk )k∈N and (b
yk )k∈N converge to x
bw as k tends to infinity. See Figure 10 for an illustration.
converges to U
bw be the light-like eigendirection of w, then
b ∈ U
For a parabolic element w ∈ Wpar , let x
⊥
b
b
b
b at x
b. See Figure 11 for an
Uw + Uw = Hx is the codimension-1 hyperplane that is tangent to Q
b. It is generated by reflections through
illustration. Let Wx 3 w be the stabilizer subgroup of x
positive roots [Dye13, Lemma 1.10]
b) = 0} ⊂ Hx .
Φx = {γ ∈ Φ+ | B(γ, x
1616
H.CHEN
CHENAND
ANDJ.-P.
J.-P.LABB
LABB
H.
ÉÉ
�w
U
α
�
�k
U
α
�k
�
Q
β�
�
x
�k
x
�k
y
β�k
�
y
Figure
Figure10.
10. Illustration
Illustrationfor
forthe
theproof
proofofofTheorem
Theorem4.2
4.2ininthe
thecase
caseofofhyperbolic
hyperbolic
elements
elementsatatstep
stepk.k.
Since the restriction kof B on Hx is �positivek semi-definite
with a radical of dimension 1, Wx is
� k∈N converge
�, they are also asymptotically
sequences (�
αk = w · α
�)k∈N and (βk = w · β)
to x
b
an irreducible
affine
Coxeter
group,
and
x
is
the
only
limit
root,
see [HLR14, Corollary 2.16].
⊥
⊥
�
�
�, in the sense that limk→∞ (�
x
− have
α
�k−1 )U∈⊥ U⊆w .U and H ⊆ U , which is not
tangent to Uwweatclaim
Furthermore,
that Φx 6⊂ Uw , otherwiseαkwe
w
x
w
w
Since the simple roots ∆ spans V and the Coxeter group
W is Lorentzian, there is a simple
possible.
�x
� at two points. We now construct two sequences of
�find
root
δ such
theΦ
[δ,
] intersects
Q
b xsegment
b ∈that
Since
x
conv(
), we
can
two positive
roots α, β ∈ Φx with opposite sign for the
�
k
�
k
�
� k∈N . Both sequences
� and
projectiveb roots
α
(β�kthat,
= wunder
σβ · δ)
converge
to x
k = w σα · δ)k∈N
coefficient
in the(�
decomposition
(3). and
Recall
the action of (wk )k∈N , the
coefficients
bk
⊥
�
�
�
�
�
�. The
are asymptotically
tangent(4)
to as
Uwk at
x
sequenceTherefore,
of segments
α
intersect
dominates
ck in Equation
tends
to infinity.
in [�
the
space,
as theQ
k , βprojective
k ] eventually
�k and
�βbkk, =
at two limit
x
which
a unimodular
projective
Uk =
b k∈N converge
b, they
sequences
(b
αk roots,
= wk ·say
α
b)k∈N
and (y
wk · determine
β)
to x
are also subspace
asymptotically
�
�
Λ(�
x
,
y
)
∈
L
.
The
two
sequences
of
limit
roots
(�
x
)
and
(�
y
)
both
converge
to
x
, and
⊥
⊥
k
k
hyp
k
k∈N
k
k∈N
b
b
b, in the sense that limk→∞ (b
αk − α
bk−1 ) ∈ Uw .
tangent to Uw at x
�w⊥ at x
�k converges to the orthogonal
�
are
asymptotically
tangent
to
U
.
By
the
continuity
of
Λ,
U
Since the simple roots ∆ spans V and the Coxeter group W is Lorentzian, there is a simple
⊥
�the
�[δ,
�w , as kQ
bw⊥⊥
btends
companion
of U
is U
U
to points.
infinity.We now construct two sequences of
b] =
root
δ such that
segment
x
intersects
at two
w , which
0
k
0
k
b k∈N and (βb = w σβ · δ)
b k∈N . Both sequences converge to x
b and
projective roots (b
αk = w σα · δ)
k
⊥
0
0
b
b
b
�k
b. The sequence
are asymptotically tangent to Uw at x
of segments [b
αk , βk ] eventually intersect Q
U
bk and y
bk , which determine a unimodular projective subspace Uk =
at two limit roots, say x
bk ) ∈ Lhyp . The two α�sequences
b, and
Λ(b
xk , y
of limit roots (b
xk )k∈N and (b
yk )k∈N
β�k both converge to x
k
α
�
⊥
β� to the orthogonal
b
b
b. By the continuity of Λ, Uk converges
are asymptotically tangent to Uw at x
�
x
� ⊥⊥
b ⊥ , which isα�U
b
bw , as k tends to infinity.
β�k�
companion of U
=U
kw
w
�k
�k
x
y
4.3. Limit directions. By Theorem 3.9, we have Lhyp ⊂ Uhyp ⊂ EV . Notably, while limit roots
arise from projective roots and projective weights, it is possible for a projective root to be a limit
direction, and space-like projective weights are all limit directions.
In this part, we prove the other inclusion of Theorem 1.2, namely that EV ⊆ Lhyp . We will
need the following two lemmas.
Lemma 4.3 (Selberg’s lemma, [Rat06, Section 7.5]). Every finitely generated subgroup G of
GL(n, C) has a torsion-free normal subgroup of finite index.
Lemma 4.4. Let G be a group acting onδ�a vector space X, and H be a subgroup of G of finite
index. Then the set of limit points of H is equal to the set of limit points of G.
Figure 11. Illustration for the proof of Theorem 4.2 in the case of parabolic eleProof. A ments
limit point
at stepofk.H is a limit point of G. Conversely, let x ∈ X be a limit point of G
arising from a base point x0 ∈ X through the sequence (gk )k∈N ∈ G. Since H is of finite index,
by passing to a subsequence if necessary, we may assume that the sequence (gk )k∈N is contained
in a single coset of H. That is, there is a sequence (hk )k∈N ∈ H and a fixed element g ∈ G such�
�] intersects Q at two points. We now construct two sequences of
root δ such that the segment [δ, x
� k∈N and (β�� = wk σβ · δ)
� k∈N . Both sequences converge to x
� and
projective roots (�
αk� = wk σα · δ)
k
�w⊥ at x
�
�. The sequence of segments [�
are asymptotically tangent to U
αk� , β�k� ] eventually intersect Q
�k and y
�k , which determine a unimodular projective subspace Uk =
at two limit roots, say x
�k ) ∈ Lhyp . The two sequences of limit roots (�
�, and
Λ(�
xk , y
xk )k∈N and (�
yk )k∈N both converge to x
⊥
�
�
�. By the continuity of Λ, Uk converges to the orthogonal
are asymptotically tangent to Uw at x
�w⊥ , which
�w⊥⊥ = U
�w , FOR
companion of U
is DIRECTIONS
U
as k LORENTZIAN
tends to infinity.
LIMIT
COXETER SYSTEMS
17
α
�
�k
U
α
�k
α
�k�
�
x
�k
x
β�k
�k
y
β�k�
β�
δ�
Figure 11. Illustration for the proof of Theorem 4.2 in the case of parabolic eleFigure 11. Illustration for the proof of Theorem 4.2 in the case of parabolic elements at step k.
ments at step k.
�
that gk = hk g for all k ∈ N. Then x is a limit point of H arising from the point g(x0 ) through the
sequence (hk )k∈N .
Theorem 4.5. The set of limit direction of a Lorentzian Coxeter system is included in Lhyp ,
EV ⊆ Lhyp .
b ∈
Proof. Assume that some x
/ Lhyp is a limit direction. By Selberg’s lemma, there exists a
subgroup of W of finite index whose only element of finite order is the identity. By Lemma 4.4,
b arises through a sequence of infinite-order elements. By the definition of
the limit direction x
b, there is infinitely many elements w ∈ W∞ such that
limit point, for any neighborhood N of x
(w · N ) ∩ N 6= ∅.
b is not in the Tits cone T . If x
b is in the interior of T , there is a neighborhood N
We claim that x
b such that (w · N ) ∩ N = ∅ for any element w ∈ W∞ , see [AB08, Exercise 2.90]. If x
b is on
of x
b is infinite, so there is an element w ∈ W∞ such that
the boundary of T , the stabilizer of x
bw ⊂ Lhyp , contradicting our assumption. Let C ∈ Σ be the cell of H containing x
b∈U
b. Since
x
b∈
x
/ Lhyp , C does not intersect Uw for any w ∈ W∞ . We now prove that (w · C) ∩ C = ∅ for any
element w ∈ W∞ .
Assume that w ∈ W∞ is an element of infinite order such that (w · C) ∩ C 6= ∅. Since H is
bw = ∅,
invariant under the action of W , we must have wk ·C = C for any integer k ∈ Z. Since C ∩ U
b
some vertices of C is not in Uw . Let v ∈ C be such a vertex. From the proof of Theorem 3.8
and 3.9, we see that the sequence (wk (v))k∈N converges to a light-like eigendirection of w. In
order for C to be invariant under the action of w, the vertex v must be a light-like eigendirection
bw ⊂ Lhyp .
of w. The element w must be hyperbolic, otherwise if w is parabolic, we have C ⊂ U
bw . Otherwise, let u ∈
bw be another vertex
Furthermore, v is the only vertex of C that is not in U
/U
b and x
b ∈ C is in the interior of the Tits
of C, then the segment [u, v] is inside the light cone Q,
cone T . Therefore, the cell C ⊂ Hv is light-like.
Let WC be the stabilizer subgroup of C. It is generated by reflections in the positive roots
ΦC = {γ ∈ Φ+ | C ⊂ Hγ }. These roots lie on the orthogonal companion of supp(C), which is
b at v. The restriction of the bilinear form B on the subspace spanned
tangent to the light cone Q
by ΦC is positive semi-definite, so WC is an affine Coxeter group. We then conclude that there is
bw0 ⊂ Lhyp , contradicting our assumption.
an element w0 ∈ Wpar such that C ∈ U
18
H. CHEN AND J.-P. LABBÉ
b ∈ C and C is open in supp(C),
We have proved that (w(C)) ∩ C = ∅ for all w ∈ W∞ . Since x
b such that (w · N ) ∩ N = ∅ for all w ∈ W∞ . Therefore, x
b can
we can find a neighborhood N of x
not be a limit direction.
4.4. Open problems on limit directions. We proved that the set EV of limit directions is
located between the set Lhyp and its closure. In fact, by Theorem 3.9 and Section 4.2, a stronger
result can be obtained:
Uhyp t EΦ ⊆ EV ⊆ Uhyp .
18
H. CHEN AND J.-P. LABBÉ
ststu
stu
sutu
αu
αs
tusu
tsu
tstsu
αt
Figure 12. The affine representation of some space-like limit directions of the
Figure
12.Coxeter
The affine
representation
of some
space-like
limitform
directions
universal
group
of rank 3. The
associated
bilinear
has cij of= the
1.1
universal
Coxeter
group
of
rank
3.
The
associated
bilinear
form
has
cij =
1.1
whenever i �= j, same as in Figure 2. The generators of the Coxeter
group
whenever
i 6=u,j,while
sameαsas
Figure 2. The generators of the Coxeter group
are s, t and
, αin
t and αu in the figure are the corresponding simple
are
s,
t
and
u,
while
α
,
α
and
αu in the figure
are theelements
corresponding
simple
s
t
roots. The dots are unimodular eigenvectors
of hyperbolic
of length
≤ 5.
roots.
The
dots
are
unimodular
eigenvectors
of
hyperbolic
elements
of
length
≤ 5.
The six eigenvectors marked with diamonds are also weights. Some unimodular
The
six eigenvectors
marked
with
diamonds arehyperbolic
also weights.
Some unimodular
eigenspaces
are labeled
by the
corresponding
elements.
The dotted
eigenspaces
are
labeled
by
the
corresponding
hyperbolic
elements.
lines show some linear dependences between limit directions verified The
with dotted
Sage.
lines show some linear dependences between limit directions verified with Sage.
Figure 12
12 shows
shows some
some unimodular
unimodular eigenvectors
eigenvectors for
for infinite
infinite order
order elements
elements for
for the
the same
same geometric
geometric
Figure
Coxeter
system
as
in
Figure
2.
Many
of
the
eigenvectors
are
not
in
the
intersections
of the
the
Coxeter system as in Figure 2. Many of the eigenvectors are not in the intersections of
Coxeter
arrangement,
but
can
be
approximated
by
intersections.
Interestingly,
the
eigenvectors
Coxeter arrangement, but can be approximated by intersections. Interestingly, the eigenvectors
seem to
to obey
obey certain
certain linear
linear dependences
dependences which
which are
are not
not present
present in
in the
the hyperplane
hyperplane arrangement.
arrangement.
seem
For
example,
the
unimodular
eigenspaces
of
the
elements
ststu,
stu,
sutu,
tusu,
tsu
and tstsu
tstsu lie
lie
For example, the unimodular eigenspaces of the elements ststu, stu, sutu, tusu, tsu and
on a hyperplane which is not a reflecting hyperplane. It would be interesting to study these linear
dependences of unimodular eigenvectors in relation with the structure of the Coxeter group.
Problem 4.6. Prove or disprove the following equalities.
(6)
(7)
EV = Uhyp � EΦ ,
EV = Uhyp .
� (see for instance
Only one of the equality may be true. In fact, in the case where EΦ = Q
LIMIT DIRECTIONS FOR LORENTZIAN COXETER SYSTEMS
19
on a hyperplane which is not a reflecting hyperplane. It would be interesting to study these linear
dependences of unimodular eigenvectors in relation with the structure of the Coxeter group.
Problem 4.6. Prove or disprove the following equalities.
(6)
(7)
EV = Uhyp t EΦ ,
EV = Uhyp .
b (see for instance
Only one of the equality may be true. In fact, in the case where EΦ = Q
Figure 5 and Figure 8 of [HLR14] and Figure 8 of the present paper), Uhyp is the union of countably many codimension-2 subspaces, while Uhyp consists of all space-like and light-like directions,
so Uhyp is a proper subset of Uhyp . Then Equation (7) would imply counterintuitively that every
non-time-like direction is a limit direction.
In the boundary ∂(Uhyp ) = Uhyp \ Uhyp , we know that the limit roots EΦ ⊂ ∂(Uhyp ) are limit
directions. Since every limit direction in Uhyp arises through a sequence of the form (wk )k∈N for
w ∈ W∞ , Equation (6) is equivalent to the following conjecture
Conjecture 4.7. Every space-like limit direction arises through a sequence of the form (wk )k∈N
for w ∈ W∞ .
To prove Equation (7), it suffices to prove that EV is closed. Let Ex be the set of limit directions
b ∈ PV . Since Ex is the set of accumulation points of W · x
b, it is
arising from a fixed base point x
b and y
b lie
clear that EV is closed and invariant under the action of W . In particular, Ex = Ey if x
b = w·y
b for some w ∈ W . However, the set of limit directions, being
in a same orbit of W , i.e. x
the infinite union
[
[
EV =
Ex =
Ex ,
b∈PV
x
b∈PV /G
x
may not be closed in general.
b ∈ PV . In
Since the set of limit roots EΦ is a minimal set of W , we have EΦ ⊆ Ex for all x
b is a time-like, light-like, projective root or a projective
Section 2, we have seen that Ex = EΦ if x
b is in
weight. Using the argument in the proof of Theorem 4.5, we can prove that Ex = EΦ if x
the Tits cone T . Define the catchment set FΦ = {b
x ∈ PV | Ex = EΦ } of limit roots.
Problem 4.8. Is FΦ a closed set?
References
[AB08] Peter Abramenko and Kenneth S. Brown, Buildings, GTM, vol. 248, Springer, New York, 2008.
[AVS93] Dmitri. V. Alekseevskij, Ernest. B. Vinberg, and Aleksandr. S. Solodovnikov, Geometry of spaces of
constant curvature, Geometry, II, Encyclopaedia Math. Sci., vol. 29, Springer, Berlin, 1993, pp. 1–138.
[Bou68] Nicolas Bourbaki, Groupes et algèbres de Lie. Chapitre 4-6, Paris: Hermann, 1968.
[Cec08] Thomas E. Cecil, Lie sphere geometry, 2 ed., Universitext, Springer, New York, 2008.
[CG00] Jean-Pierre Conze and Yves Guivarc’h, Limit sets of groups of linear transformations, Sankhyā Ser. A
62 (2000), no. 3, 367–385.
[CL14] Hao Chen and Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell Packings, preprint,
arXiv:1310.8608 (2014), 28 pp.
[DHR13] Matthew Dyer, Christophe Hohlweg, and Vivien Ripoll, Imaginary cones and limit roots of infinite
Coxeter groups, preprint, arXiv:1303.6710 (March 2013), 63 pp.
[Dye11] Matthew Dyer, On the weak order of Coxeter groups, preprint, arXiv:abs/1108.5557 (August 2011), 37
pp.
[Dye13]
, Imaginary cone and reflection subgroups of Coxeter groups, preprint, arXiv:1210.5206 (2013),
89 pp.
[HLR14] Christophe Hohlweg, Jean-Philippe Labbé, and Vivien Ripoll, Asymptotical behaviour of roots of infinite
Coxeter groups, Canad. J. Math. 66 (2014), no. 2, 323–353.
[HPR13] Christophe Hohlweg, Jean-Philippe Préaux, and Vivien Ripoll, On the limit set of root systems of Coxeter
groups and Kleinian groups, preprint, arXiv:1305.0052 (July 2013), 24 pp.
[Hum92] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics. 29 (Cambridge University Press), 1992.
[Kra09] Daan Krammer, The conjugacy problem for Coxeter groups, Group. Geom. Dynam. 3 (2009), no. 1,
71–171.
20
H. CHEN AND J.-P. LABBÉ
[Lab13]
Jean-Philippe Labbé, Polyhedral Combinatorics of Coxeter Groups, Ph.D. thesis, Freie Universität Berlin,
July 2013, available at http://www.diss.fu-berlin.de/diss/receive/FUDISS thesis 000000094753,
pp. xvi+103.
[LP13]
Thomas Lam and Pavlo Pylyavskyy, Total positivity for loop groups II: Chevalley generators, Transform.
Groups 18 (2013), no. 1, 179–231.
[LR14] Jean-Philippe Labbé and Vivien Ripoll, Implementation of geometric Coxeter systems, Sage Days 57,
The Sage Development Team, 2014, http://wiki.sagemath.org/days57.
[LT13]
Thomas Lam and Anne Thomas, Infinite reduced words and the Tits boundary of a Coxeter group,
preprint, arXiv:1301.0873 (January 2013), 26 pp.
[Mar07] Albert Marden, Outer circles, Cambridge University Press, Cambridge, 2007.
[Max82] George Maxwell, Sphere packings and hyperbolic reflection groups, J. Algebra 79 (1982), no. 1, 78–97.
[Pil06]
Annette Pilkington, Convex geometries on root systems, Comm. Algebra 34 (2006), no. 9, 3183–3202.
[Rat06] John G. Ratcliffe, Foundations of hyperbolic manifolds, second ed., GTM, vol. 149, Springer, New York,
2006.
[Rie58] Marcel Riesz, Clifford numbers and spinors (Chapters I–IV), Lecture Series, No. 38. The Institute for
Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Md., 1958.
[S+ 14]
William A. Stein et al., Sage Mathematics Software (Version 6.1.1), The Sage Development Team, 2014,
http://www.sagemath.org.
[SR13]
Igor R. Shafarevich and Alexey O. Remizov, Linear algebra and geometry, Springer, Heidelberg, 2013.
[Vin71] Ernest B. Vinberg, Discrete linear groups that are generated by reflections, Izv. Akad. Nauk SSSR. Ser.
Mat. 35 (1971), 1072–1112.
(H. Chen) Freie Universität Berlin, Institut für Mathematik, Arnimallee 2, 14195 Berlin, Deutschland
E-mail address: [email protected]
URL: http://page.mi.fu-berlin.de/haochen
(J.-P. Labbé) Freie Universität Berlin, Institut für Mathematik, Arnimallee 2, 14195 Berlin, Deutschland
E-mail address: [email protected]
URL: http://page.mi.fu-berlin.de/labbe