Are all Coxeter groups virtually free or virtually surface groups?

Question

From Surface subgroups of Coxeter and Artin groups (Gordon, Long and Reid, 2003) DOI link, 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?

Answer 1

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).