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

Caprace, Pierre-Emmanuel

doi:10.2140/agt.2006.6.1987

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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

DOI: 10.2140/agt.2006.6.1987

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

Volume 6, issue 4 (2006)