Conjugacy of $2$–spherical subgroups of Coxeter groups and parallel walls

Abstract

Let $\left(W,S\right)$ be a Coxeter system of finite rank (ie $\left|S\right|$ is finite) and let $\mathcal{A}$ be the associated Coxeter (or Davis) complex. We study chains of pairwise parallel walls in $\mathcal{A}$ using Tits’ bilinear form associated to the standard root system of $\left(W,S\right)$. As an application, we prove the strong parallel wall conjecture of G Niblo and L Reeves [J Group Theory 6 (2003) 399–413]. This allows to prove finiteness of the number of conjugacy classes of certain one-ended subgroups of $W$, which yields in turn the determination of all co-Hopfian Coxeter groups of $2$–spherical type.