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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 (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 [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
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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