It is known after Stallings that a group can have 0, 1, 2 or infinitely many ends. Are there known results on the space of ends of a Coxeter group? 



Ends of Coxeter Groups
The following book has a wealth of material on this topic:



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> *The geometry and topology of Coxeter groups* by Michael Davis.
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By way of example, here is one result from the book:



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> **Theorem 8.7.1** Let $(W,S)$ be a Coxeter System:
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> 1. $W$ is one-ended if and only $H\_c^1(\Sigma)=0$.
> 2. $W$ has two ends if and only $H\_c^1(\Sigma)\cong\mathbb{Z}$.
> 3. $W$ has infinitely many ends if and only if $H\_c^1(\Sigma)$ has infinite rank.
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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.


