The FAn Conjecture for Coxeter groups

Barnhill, Angela Kubena

doi:10.2140/agt.2006.6.2117

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The $\mathrm{FA}_n$ Conjecture for Coxeter groups

Angela Kubena Barnhill

Algebraic & Geometric Topology 6 (2006) 2117–2150

DOI: 10.2140/agt.2006.6.2117

arXiv: math.GR/0509439

Abstract

We study global fixed points for actions of Coxeter groups on nonpositively curved singular spaces. In particular, we consider property ${FA}_{n}$ , an analogue of Serre’s property FA for actions on $CAT (0)$ complexes. Property ${FA}_{n}$ has implications for irreducible representations and complex of groups decompositions. In this paper, we give a specific condition on Coxeter presentations that implies ${FA}_{n}$ and show that this condition is in fact equivalent to ${FA}_{n}$ for $n = 1$ and 2. As part of the proof, we compute the Gersten–Stallings angles between special subgroups of Coxeter groups.

Keywords

Coxeter group, fixed point, nonpositive curvature, triangle of groups, complex of groups

Mathematical Subject Classification 2000

Primary: 20F65

Secondary: 20F55

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Publication

Received: 25 September 2005

Accepted: 6 March 2006

Published: 19 November 2006

Authors

Angela Kubena Barnhill
Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, Ohio 43210 USA

Volume 6, issue 5 (2006)