#### Volume 6, issue 5 (2006)

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

### Angela Kubena Barnhill

Algebraic & Geometric Topology 6 (2006) 2117–2150
 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 $CAT\left(0\right)$ 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
Primary: 20F65
Secondary: 20F55