From *Surface subgroups of Coxeter and Artin groups* (Gordon, Long and Reid, 2003) [DOI link](https://doi.org/10.1016/j.jpaa.2003.10.011), we can read that (Theorem 1.1) a Coxeter group is either virtually free or contains a surface group ($\pi\_1$ of a closed orientable surface of genus $\ge 1$).


My question is:
Are all Coxeter groups virtually free or virtually surface groups?



Are all Coxeter groups virtually free or virtually surface groups?
When asking a question about Coxeter groups, it might be useful to focus on right-angled Coxeter groups first: usually, they are easier to handle. Below are a few criteria you can play with in order to create various examples of (right-angled) Coxeter groups.


Let $\Gamma$ be a finite simplicial graph. The right-angled Coxeter group $C(\Gamma)$ is:


* finite if and only if $\Gamma$ is complete;
* virtually free if and only if $\Gamma$ is chordal (i.e. it does not contain an induced cycle of length $\geq 4$);
* virtually abelian of rank $n \geq 1$ if and only if $\Gamma$ decomposes as a join of a complete graph and $n$ copies of $K\_2^{\mathrm{opp}}$ (= two isolated vertices);
* virtually a hyperbolic surface group if and only if $\Gamma$ decomposes as a join of a complete graph and a cycle of length $\geq 5$;
* hyperbolic if and only if $\Gamma$ is square-free (i.e. it does not contain an induced cycle of length $4$);
* a free product if and only if $\Gamma$ is disconnected;
* multi-ended if and only if $\Gamma$ contains a separating complete subgraph (possibly empty).


