#### Volume 6, issue 4 (2006)

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

### Pierre-Emmanuel Caprace

Algebraic & Geometric Topology 6 (2006) 1987–2029
 arXiv: math.GR/0508057
##### Abstract

Let $\left(W,S\right)$ be a Coxeter system of finite rank (ie $|S|$ is finite) and let $\mathsc{A}$ be the associated Coxeter (or Davis) complex. We study chains of pairwise parallel walls in $\mathsc{A}$ 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.

##### Keywords
Coxeter group, conjugacy class, Hopfian group, hyperbolic triangle, parallel walls
##### Mathematical Subject Classification 2000
Primary: 20F5
Secondary: 20F65, 20F67, 51F15
##### Publication
Received: 2 August 2005
Revised: 31 August 2006
Accepted: 4 October 2006
Published: 14 November 2006
##### Authors
 Pierre-Emmanuel Caprace Département de Mathématiques Université Libre de Bruxelles, CP216 Bd du Triomphe 1050 Bruxelles Belgium