Dihedral subgroup of a infinite Coxeter group

Question

I have seen a conclusion that every infinite Coxeter group contain an infinite dihedral subgroup, but I have no idea how to prove it. Could anyone give me some hint?

Answer 1

1. First of all, this property is false if you allow infinitely generated Coxeter groups, for instance, the direct sum of infinitely many ${\mathbb Z}_2$'s is an infinite Coxeter group without infinite dihedral subgroups.

2. Assuming that $W$ is a finitely generated infinite Coxeter group, always contains two reflections in parallel hyperplanes (with respect to, say, Tits-Bourbaki representation). For simplicity, I will assume that $W$ is irreducible and of "hyperbolic type." Let $W\to GL(V)$ be the Tits-Bourbaki representation of $W$. Let $\Omega\subset V$ be the (open) Tits cone of the action of $W$ on $V$. Each reflection $\tau\in W$ fixes a linear hyperplane $H_\tau$ in $V$. Two reflections $\tau, \sigma$ are said to have parallel walls if $$ H_\tau \cap H_\sigma \cap \Omega=\emptyset. $$ One verifies that such reflections $\tau,\sigma$ generate an infinite dihedral subgroup. The existence of reflections with parallel walls in each infinite Coxeter group of finite rank (Parallel Wall Property) was first stated and proven in

Davis, M.W., Shapiro, M.D.: Coxeter groups are automatic. Ohio Slate University (1991) (preprint).

Aparently, their proof was incomplete (do not ask me what was missing), the gap was fixed in:

Brink, Brigitte; Howlett, Robert B., A finiteness property and an automatic structure for Coxeter groups, Math. Ann. 296, No. 1, 179-190 (1993). ZBL0793.20036.