Thickness, relative hyperbolicity, and randomness in Coxeter groups

  title={Thickness, relative hyperbolicity, and randomness in Coxeter groups},
  author={Jason A. Behrstock and Mark F. Hagen and Alessandro Sisto and Pierre‐Emmanuel Caprace},
  journal={Algebraic \& Geometric Topology},
For right-angled Coxeter groups $W_{\Gamma}$, we obtain a condition on $\Gamma$ that is necessary and sufficient to ensure that $W_{\Gamma}$ is thick and thus not relatively hyperbolic. We show that Coxeter groups which are not thick all admit canonical minimal relatively hyperbolic structures; further, we show that in such a structure, the peripheral subgroups are both parabolic (in the Coxeter group-theoretic sense) and strongly algebraically thick. We exhibit a polynomial-time algorithm that… 

Figures and Tables from this paper

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

Algebraic characterisation of relatively hyperbolic special groups

  • A. Genevois
  • Mathematics
    Israel Journal of Mathematics
  • 2021
This article is dedicated to the characterisation of the relative hyperbolicity of Haglund and Wise’s special groups. More precisely, we introduce a new combinatorial formalism to study (virtually)

Generic free subgroups and statistical hyperbolicity

This paper studies the generic behavior of $k$-tuple elements for $k\ge 2$ in a proper group action with contracting elements, with applications towards relatively hyperbolic groups, CAT(0) groups

Divergence, thickness and hypergraph index for general Coxeter groups

. We study divergence and thickness for general Coxeter groups W . We first characterise linear divergence, and show that if W has superlinear divergence then its divergence is at least quadratic. We

A remark on thickness of free-by-cyclic groups

  • M. Hagen
  • Mathematics
    Illinois Journal of Mathematics
  • 2019
Let $F$ be a free group of positive, finite rank and let $\Phi\in Aut(F)$ be a polynomial-growth automorphism. Then $F\rtimes_\Phi\mathbb Z$ is strongly thick of order $\eta$, where $\eta$ is the

Divergence of CAT(0) Cube Complexes and Coxeter Groups

We provide geometric conditions on a pair of hyperplanes of a CAT(0) cube complex that imply divergence bounds for the cube complex. As an application, we classify all right-angled Coxeter groups

Algebraic characterisations of negatively-curved special groups and applications to graph braid groups

In this paper, we introduce a combinatorial formalism to study (virtually) special groups, introduced by Haglund and Wise. As a first application, we recover a result due to Caprace and Haglund: if

Genericity of contracting elements in groups

We establish that, for statistically convex-cocompact actions, contracting elements are exponentially generic in counting measure. We obtain as corollaries results on the exponential genericity for

Subgroups of right-angled Coxeter groups via Stallings-like techniques

We associate a cube complex to any given finitely generated subgroup of a right-angled Coxeter group, called the completion of the subgroup. A completion characterizes many properties of the subgroup

Malnormality and join-free subgroups in right-angled Coxeter groups

In this paper, we prove that all finitely generated malnormal subgroups of one-ended right-angled Coxeter groups are strongly quasiconvex and they are in particular quasiconvex when the ambient



Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

We study the geometry of non-relatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is

Relatively hyperbolic Groups

This paper defines the boundary of a relatively hyperbolic group, and shows that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary, and generalizes a result of Tukia for geometRically finite kleinian groups.

Buildings with isolated subspaces and relatively hyperbolic Coxeter groups

Let $(W, S)$ be a Coxeter system. We give necessary and sufficient conditions on the Coxeter diagram of $(W, S)$ for $W$ to be relatively hyperbolic with respect to a collection of finitely generated

Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries

Let G be a group acting geometrically on a CAT(0) cube complex X. We prove first that G is hyperbolic relative to the collection P of subgroups if and only if the simplicial boundary of X is the

Automorphisms of Graph-Universal Coxeter Groups☆

Coxeter groups play an important role in various areas of mathematics, such as combinatorics, geometry, and Lie theory. It is somewhat surprising that there haven’t been many attempts to study their

Coarse and synthetic Weil–Petersson geometry: quasi-flats, geodesics and relative hyperbolicity

We analyze the coarse geometry of the Weil-Petersson metric on Teichm¨ uller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the

Random groups arising as graph products

In this paper we study the hyperbolicity properties of a class of random groups arising as graph products associated to random graphs. Recall, that the construction of a graph product is a

Divergence, thick groups, and short conjugators

In this paper we explore relationships between divergence and thick groups, and with the same techniques we estimate lengths of shortest conjugators. We produce examples, for every positive integer

Global Structural Properties of Random Graphs

We study two global structural properties of a graph $\Gamma$, denoted AS and CFS, which arise in a natural way from geometric group theory. We study these properties in the Erdos--Renyi random graph

Quasi-isometric classification of graph manifold groups

We show that the fundamental groups of any two closed irreducible nongeometric graph manifolds are quasi-isometric. We also classify the quasi-isometry types of fundamental groups of graph manifolds