Conjugacy of 2-spherical subgroups of Coxeter groups and parallel walls

Let (W, S) be a Coxeter system of finite rank (ie |S| is finite) and let A be the associated Coxeter (or Davis) complex. We study chains of pairwise parallel walls in A using Tits' bilinear form associated to the standard root system of (W, S). As an application, we prove the strong parallel wall conjecture of G Niblo and L Reeves [18]. 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.