The following book has a wealth of material on this topic: > *The geometry and topology of Coxeter groups* by Michael Davis. By way of example, here is one result from the book: > **Theorem 8.7.1** Let $(W,S)$ be a Coxeter System: > > 1. $W$ is one-ended if and only $H_c^1(\Sigma)=0$. > > 2. $W$ has two ends if and only $H_c^1(\Sigma)\cong\mathbb{Z}$. > > 1. $W$ has infinitely many ends if and only if $H_c^1(\Sigma)$ has infinite rank. Here $\Sigma$ is the cell complex associated to $(W,S)$. I have an e-copy of the book - email me if you want a copy.