Right-angled Coxeter Groups Research Articles

Automorphisms of Contact Graphs of CAT(0) Cube Complexes

Abstract We show that, under weak assumptions, the automorphism group of a $\textrm{CAT(0)}$ cube complex $X$ coincides with the automorphism group of Hagen’s contact graph $\mathcal{C}(X)$. The result holds, in particular, for universal covers of Salvetti complexes, where it provides an analogue of Ivanov’s theorem on curve graphs of non-sporadic surfaces. This highlights a contrast between contact graphs and Kim–Koberda extension graphs, which have much larger automorphism group. We also study contact graphs associated with Davis complexes of right-angled Coxeter groups. We show that these contact graphs are less well behaved and describe exactly when they have more automorphisms than the universal cover of the Davis complex.

Growth series for graphs

Given a graph Γ, one can associate a right-angled Coxeter group W and a cube complex Σ on which W acts. By identifying W with the vertex set of Σ, one obtains a growth series for W defined as W(t)= ∑w∈Wtl(w), where l(w) denotes the minimum length of an edge path in Σ from the vertex 1 to the vertex w. The series W(t) is known to be a rational function. We compute some examples and investigate the poles and zeros of this function.

Proper affine actions for right-angled Coxeter groups

For any right-angled Coxeter group $\Gamma$ on $k$ generators, we construct proper actions of $\Gamma$ on $\mathrm{O}(p,q+1)$ by right and left multiplication, and on the Lie algebra $\mathfrak{o}(p,q+1)$ by affine transformations, for some $p,q\in\mathbb N$ with $p+q+1=k$. As a consequence, any virtually special group admits proper affine actions on some $\mathbb R^n$: this includes e.g. surface groups, hyperbolic 3-manifold groups, examples of word hyperbolic groups of arbitrarily large virtual cohomological dimension, etc. We also study some examples in cohomological dimension two and four, for which the dimension of the affine space may be substantially reduced.

Random coxeter groups

Much is known about random right-angled Coxeter groups (i.e., right-angled Coxeter groups whose defining graphs are random graphs under the Erdös–Rényi model). In this paper, we extend this model to study random general Coxeter groups and give some results about random Coxeter groups, including some information about the homology of the nerve of a random Coxeter group and results about when random Coxeter groups are [Formula: see text]-hyperbolic and when they have the FC-type property.

Conjugacy classes and automorphisms of twin groups

Abstract The twin group T n {T_{n}} is a right-angled Coxeter group generated by n - 1 {n-1} involutions, and the pure twin group PT n {\mathrm{PT}_{n}} is the kernel of the natural surjection from T n {T_{n}} onto the symmetric group on n symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conjugacy classes of involutions in T n {T_{n}} , which, quite interestingly, is related to the well-known Fibonacci sequence. We also derive a recursive formula for the number of z-classes of involutions in T n {T_{n}} . We give a new proof of the structure of Aut ⁡ ( T n ) {\operatorname{Aut}(T_{n})} for n ≥ 3 {n\geq 3} , and show that T n {T_{n}} is isomorphic to a subgroup of Aut ⁡ ( PT n ) {\operatorname{Aut}(\mathrm{PT}_{n})} for n ≥ 4 {n\geq 4} . Finally, we construct a representation of T n {T_{n}} to Aut ⁡ ( F n ) {\operatorname{Aut}(F_{n})} for n ≥ 2 {n\geq 2} .

Counting problems in graph products and relatively hyperbolic groups

We study properties of generic elements of groups of isometries of hyperbolic spaces. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e., they have full density with respect to counting in balls for the word metric in the Cayley graph) and translation length grows linearly. We provide applications to a large class of relatively hyperbolic groups and graph products, including all right-angled Artin groups and right-angled Coxeter groups.

Absence of Cartan subalgebras for right-angled Hecke von Neumann algebras

For a right-angled Coxeter system $(W,S)$ and $q>0$, let $\mathcal{M}_q$ be the associated Hecke von Neumann algebra, which is generated by self-adjoint operators $T_s, s \in S$ satisfying the Hecke relation $(\sqrt{q}\: T_s - q) (\sqrt{q} \: T_s + 1) = 0$ as well as suitable commutation relations. Under the assumption that $(W, S)$ is irreducible and $\vert S \vert \geq 3$ it was proved by Garncarek that $\mathcal{M}_q$ is a factor (of type II$_1$) for a range $q \in [\rho, \rho^{-1}]$ and otherwise $\mathcal{M}_q$ is the direct sum of a II$_1$-factor and $\mathbb{C}$. In this paper we prove (under the same natural conditions as Garncarek) that $\mathcal{M}_q$ is non-injective, that it has the weak-$\ast$ completely contractive approximation property and that it has the Haagerup property. In the hyperbolic factorial case $\mathcal{M}_q$ is a strongly solid algebra and consequently $\mathcal{M}_q$ cannot have a Cartan subalgebra. In the general case $\mathcal{M}_q$ need not be strongly solid. However, we give examples of non-hyperbolic right-angled Coxeter groups such that $\mathcal{M}_q$ does not possess a Cartan subalgebra.

Surface group amalgams that (don't) act on 3-manifolds

We determine which amalgamated products of surface groups identified over multiples of simple closed curves are not fundamental groups of $3$-manifolds. We prove each surface amalgam considered is virtually the fundamental group of a $3$-manifold. We prove that each such surface group amalgam is abstractly commensurable to a right-angled Coxeter group from a related family. In an appendix, we determine the quasi-isometry classes among these surface amalgams and their related right-angled Coxeter groups.

Spherical and geodesic growth rates of right-angled Coxeter and Artin groups are Perron numbers

We prove that for any infinite right-angled Coxeter or Artin group, its spherical and geodesic growth rates (with respect to the standard generating set) either take values in the set of Perron numbers, or equal 1. Also, we compute the average number of geodesics representing an element of given word-length in such groups.

The relative hyperbolicity and manifold structure of certain right-angled Coxeter groups

In this paper, we study the manifold structure and the relative hyperbolic structure of right-angled Coxeter groups with planar nerves. We then apply our results to the quasi-isometry problem for this class of right-angled Coxeter groups.

Aut-Invariant Word Norm on Right-Angled Artin and Right-Angled Coxeter Groups

We prove that the Aut-invariant word norm on right-angled Artin and right-angled Coxeter groups is unbounded (except in few special cases). To prove unboundedness, we exhibit certain characteristic subgroups. This allows us to find unbounded quasi-morphisms which are Lipschitz with respect to the Aut-invariant word norm.

Coxeter Groups and Abstract Elementary Classes: The Right-Angled Case

We study classes of right-angled Coxeter groups with respect to the strong submodel relation of a parabolic subgroup. We show that the class of all right-angled Coxeter groups is not smooth and establish some general combinatorial criteria for such classes to be abstract elementary classes (AECs), for them to be finitary, and for them to be tame. We further prove two combinatorial conditions ensuring the strong rigidity of a right-angled Coxeter group of arbitrary rank. The combination of these results translates into a machinery to build concrete examples of AECs satisfying given model-theoretic properties. We exhibit the power of our method by constructing three concrete examples of finitary classes. We show that the first and third classes are nonhomogeneous and that the last two are tame, uncountably categorical, and axiomatizable by a single Lω1,ω-sentence. We also observe that the isomorphism relation of any countable complete first-order theory is κ-Borel reducible (in the sense of generalized descriptive set theory) to the isomorphism relation of the theory of right-angled Coxeter groups whose Coxeter graph is an infinite random graph.

Right-angled Coxeter groups, universal graphs, and Eulerian polynomials

We define a class of representations for any right-angled Coxeter group R and its Hecke algebra H(R). The group R and the algebra H(R) act on any Bruhat interval of any Coxeter system (W,S), once given a suitable function from the set of Coxeter generators of R to the power set of S. The existence of such a function is related to the problem of universality of a graph G2 constructed from the unlabeled Coxeter graph G of (W,S). When G=Pn is a path with n vertices, we conjecture that G2 is n-universal; this property is equivalent to the existence of an action of R and H(R) on the Bruhat intervals of the symmetric group Sn+1, for all right-angled groups with n generators. We prove that (Pn)2 is n-universal for forests. Eulerian polynomials arise as characters of our representations, when the Coxeter graph of R is a path. We also give a formula for the toric h-polynomial of any lower Bruhat interval in a universal Coxeter group Un, using results on the Kazhdan–Lusztig basis of H(Un).

On the coarse geometry of certain right-angled Coxeter groups

Let $\Gamma$ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph $\Gamma$ is $\mathcal{CFS}$, we prove that the right-angled Coxeter group $G_\Gamma$ is virtually a Seifert manifold group or virtually a graph manifold group and we give a complete quasi-isometry classification of these such groups. Otherwise, we prove that $G_\Gamma$ is hyperbolic relative to a collection of $\mathcal{CFS}$ right-angled Coxeter subgroups of $G_\Gamma$. Consequently, the divergence of $G_\Gamma$ is linear, or quadratic, or exponential. We also generalize right-angled Coxeter groups which are virtually graph manifold groups to certain high dimensional right-angled Coxeter groups (our families exist in every dimension) and study the coarse geometry of this collection. We prove that strongly quasiconvex torsion free infinite index subgroups in certain graph of groups are free and we apply this result to our right-angled Coxeter groups.

Outer automorphism groups of right-angled Coxeter groups are either large or virtually abelian

We generalize the notion of a separating intersection of links (SIL) to give necessary and sufficient criteria on the defining graph Γ \Gamma of a right-angled Coxeter group W Γ W_\Gamma so that its outer automorphism group is large: that is, it contains a finite index subgroup that admits the free group F 2 F_2 as a quotient. When Out ⁡ ( W Γ ) \operatorname {Out}(W_\Gamma ) is not large, we show it is virtually abelian. We also show that the same dichotomy holds for the outer automorphism groups of graph products of finite abelian groups. As a consequence, these groups have property (T) if and only if they are finite or equivalently Γ \Gamma contains no SIL.

Subgroup growth of right‐angled Artin and Coxeter groups

We determine the factorial growth rate of the number of finite index subgroups of right-angled Artin groups as a function of the index. This turns out to depend solely on the independence number of the defining graph. We also make a conjecture for right-angled Coxeter groups and prove that it holds in a limited setting.

VIRTUALLY FIBERING RIGHT-ANGLED COXETER GROUPS

AbstractWe show that certain right-angled Coxeter groups have finite index subgroups that quotient to$\mathbb{Z}$with finitely generated kernels. The proof uses Bestvina–Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails. Among the successful examples are the right-angled reflection groups in$\mathbb{H}^{4}$with fundamental domain the 120-cell or the 24-cell.

Quasi-isometric rigidity of a class of right-angled Coxeter groups

We establish quasi-isometric rigidity for a class of right-angled Coxeter groups. Let Γ 1 \Gamma _1 , Γ 2 \Gamma _2 be joins of finite thick generalized m m -gons with m ∈ { 3 , 4 , 6 , 8 } m\in \{3,4,6,8\} . We show that the corresponding right-angled Coxeter groups are quasi-isometric if and only if Γ 1 \Gamma _1 , Γ 2 \Gamma _2 are isomorphic. We also give a construction of commensurable right-angled Coxeter groups.

Subgroup Growth of Virtually Cyclic Right-Angled Coxeter Groups and Their Free Products

We determine the asymptotic number of index n subgroups in virtually cyclic Coxeter groups and their free products as n → ∞.

An infinite sequence of ideal hyperbolic Coxeter 4-polytopes and Perron numbers

In [7], Kellerhals and Perren conjectured that the growth rates of cocompact hyperbolic Coxeter groups are Perron numbers. By results of Floyd, Parry, Kolpakov, Nonaka-Kellerhals, Komori and the author [1], [3], [8], [10], [12], [13], [21], [22], the growth rates of 2- and 3-dimensional hyperbolic Coxeter groups are always Perron numbers. Kolpakov and Talambutsa showed that the growth rates of right-angled Coxeter groups are Perron numbers [9]. For certain families of 4-dimensional cocompact hyperbolic Coxeter groups, the conjecture holds as well (see [7], [19] and also [23]). In this paper, we construct an infinite sequence of ideal non-simple hyperbolic Coxeter 4-polytopes giving rise to growth rates which are distinct Perron numbers. This is the first explicit example of an infinite family of non-compact finite volume Coxeter polytopes in hyperbolic 4-space whose growth rates are of the conjectured arithmetic nature as well.