When is a right-angled Coxeter group one-ended?

Question

Let $\Gamma$ be a simplicial graph (ie. without multiple edes nor loops). We define the associated right-angled Artin group $A(\Gamma)$ by the presentation $$\langle v \in V(\Gamma) \mid [u,v]=1 \ \text{if} \ (u,v) \in E(\Gamma) \rangle,$$ and the associated right-angled Coxeter group $C(\Gamma)$ by $$\langle v \in V(\Gamma) \mid u^2=1, [u,v]=1 \ \text{if} \ (u,v) \in E(\Gamma) \rangle,$$ where $V(\Gamma)$ and $E(\Gamma)$ denote respectively the set of vertices and edges of $\Gamma$.

From the graph $\Gamma$, it is easy to deduce the number of ends of $A(\Gamma)$. Indeed, $$\text{number of ends of} \ A(\Gamma)= \left\{ \begin{array}{cl} 0 & \text{if} \ \Gamma \ \text{is empty} \\ 2 & \text{if} \ \Gamma \ \text{is a point} \\ + \infty & \text{if} \ \Gamma \ \text{is not connected} \\ 1 & \text{otherwise} \end{array} \right..$$

The situation seems to be more complicated for $C(\Gamma)$. For example, $\Gamma$ may be connected when $C(\Gamma)$ is virtually free (and so when $C(\Gamma)$ has infinitely many ends). So my question is:

Is it possible to deduce from the graph $\Gamma$ when the associated right-angled Coxeter group $C(\Gamma)$ is one-ended?

Answer 1

For everything you want to know about ends of Coxeter groups see Section 8.7.3 of Mike Davis' book "The Geometry and Topology of Coxeter Groups".

Answer 2

As mentionned by Lee Mosher, Section 8.7 of Davis' book completely answers the question. Below, I apply these results to the more specific case of right-angled Coxeter groups.

Let $\Gamma$ be a graph. The nerve of the associated Coxeter system is just the flag completion $f(\Gamma)$ of $\Gamma$: if there is the boundary of a triangle in $\Gamma$, glue a triangle along it, defining a 2-complex $\Gamma_2$; if there is the boundary of a 3-simplex in $\Gamma_2$, glue a 3-simplex along it, defining a 3-complex $\Gamma_3$; etc. Then, $f(\Gamma)= \bigcup\limits_{n \geq 2} \Gamma_n$. Whatever, the 1-squeleton of the nerve is just $\Gamma$.

Now, let $\Lambda \leq \Gamma$ be an induced subgraph. The associated right-angled Coxeter group $C(\Lambda)$ is finite if and only if $\Lambda$ is a complete graph, ie. a clique of $\Gamma$. Indeed, it there exist two non-adjacent vertices in $\Lambda$, then they generate a subgroup isomorphic to the infinite dihedral group $D_{\infty}$. Thus, the spherical sets correspond to cliques in $\Gamma$.

From the two remarks above, we see that the 1-squeletons of puncture nerves correspond to $\Gamma$ minus a clique.

Finally, Theorems 8.7.2 and 8.7.3 in Davis' book give:

$$\text{number of ends of} \ C(\Gamma)= \left\{ \begin{array}{cl} 0 & \text{if} \ \Gamma \ \text{is a clique} \\ 1 & \text{if} \ \Gamma \ \text{has no separating clique} \\ 2 & \text{if} \ \Gamma= K \ast V_2 \\ + \infty & \text{otherwise} \end{array} \right..$$

By $\Gamma= K \ast V_2$, I mean that $\Gamma$ is the join of a complete graph $K$ with the disjoint union $V_2$ of two vertices. A clique $\Lambda \leq \Gamma$ is separating if its complement in $\Gamma$ is not connected.