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

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.