Related papers: Automorphisms of odd Coxeter groups

In this paper we exhibit two infinite families of trees $\{T^1_n\}_{n \geq 17}$ and $\{T^2_n\}_{n \geq 17}$ on $n$ vertices, such that $T^1_n$ and $T^2_n$ are non-isomorphic, co-spectral, and the right-angled Coxeter groups (RACGs) based on…

We define ``star reducible'' Coxeter groups to be those Coxeter groups for which every fully commutative element (in the sense of Stembridge) is equivalent to a product of commuting generators by a sequence of length-decreasing star…

It has long been known that a vertex-transitive graph $\Gamma$ is isomorphic to a double coset graph $\text{Cos}(G,H,S)$ of a transitive group $G\le\text{Aut}(\Gamma)$, a vertex stabilizer $H\le G$, and some subset $S\subseteq G$. We show…

Given a finite and connected two-dimensional $CW$-complex $K$ with fundamental group $\Pi$ and second integer cohomology group $H^2(K;\mathbb{Z})$ finite of odd order, we prove that: (1) for each local integer coefficient system…

In this article we provide a new finite class of elements in any Coxeter system (W,S) called low elements. They are defined from Brink and Howlett's small roots, which are strongly linked to the automatic structure of (W,S). Our first main…

Let $W$ be a Coxeter group and $r\in W$ a reflection. If the group of order 2 generated by $r$ is the intersection of all the maximal finite subgroups of $W$ that contain it, then any isomorphism from $W$ to a Coxeter group $W'$ must take…

For each positive integer $k$ we present an example of Coxeter system $(G_k,S_k)$ such that $G_k$ is a word-hyperbolic Coxeter group, for any two generating reflections $s,t\in S_k$ the product $st$ has finite order, and the Coxeter graph…

For a Coxeter system $(W,S)$ let $a_n^{(W,S)}$ be the cardinality of the sphere of radius $n$ in the Cayley graph of $W$ with respect to the standard generating set $S$. It is shown that, if $(W,S)\preceq(W',S')$ then $a_n^{(W,S)}\leq…

Results are obtained concerning the roots of asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a…

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…

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 subgroups. The peripheral subgroups are…

A right-angled Coxeter group is a group with a given set of generators of order two, subject only to the relations that certain pairs of the generators commute. Various papers have shown how homological properties of the Coxeter group are…

Let $W$ be a right-angled Coxeter group corresponding to a finite non-discrete graph $\mathcal{G}$ with at least $3$ vertices. Our main theorem says that $\mathcal{G}^c$ is connected if and only if for any infinite index quasiconvex…

We consider presentations that were derived in \cite{BaumeisterNeaimeRees} for the interval groups associated with proper quasi-Coxeter elements of the Coxeter group $W(D_n)$. We use combinatorial methods to derive alternative presentations…

Let $G$ be a finite group and let $S$ be an inverse-closed subset of $G$ not containing the identity. The Cayley graph $\mathrm{Cay}(G,S)$ has vertex set $G$, where two vertices $x$ and $y$ are adjacent if and only if $x^{-1}y \in S$.…

Given a finite group $G$ and a conjugacy class of involutions $X$ of $G$, we define the commuting involution graph $\mathcal{C}(G,X)$ to be the graph with vertex set $X$ and $x,y \in X$ adjacent if and only if $x \neq y$ and $xy =yx$. In…

We prove that a "random" free group outer automorphism is an ageometric fully irreducible outer automorphism whose ideal Whitehead graph is a union of triangles. In particular, we show that its attracting (and repelling) tree is a…

Let W be an infinite Coxeter group. We initiate the study of the set E of limit points of "normalized" positive roots (representing the directions of the roots) of W. We show that E is contained in the isotropic cone of the bilinear form B…

In the spirit of peripheral subgroups in relatively hyperbolic groups, we exhibit a simple class of quasi-isometrically rigid subgroups in graph products of finite groups, which we call eccentric subgroups. As an application, we prove that,…

We characterize those countable rooted trees whose full automorphism group has uncountable strong cofinality or contains an open subgroup with ample generics.