Cubulated hyperbolic groups admit Anosov representations

TL;DR

This paper proves that hyperbolic groups acting on CAT(0) cube complexes can be represented via projective Anosov representations, extending the understanding of their geometric and algebraic properties.

Contribution

It establishes that hyperbolic groups acting on CAT(0) cube complexes admit projective Anosov representations, especially for subgroups of right-angled Coxeter groups.

Findings

Hyperbolic groups acting on CAT(0) cube complexes admit Anosov representations.

Generic reflection representations of right-angled Coxeter groups restrict to Anosov representations.

The result links geometric actions on cube complexes with algebraic representations in SL(d, R).

Abstract

We prove that any hyperbolic group acting properly discontinuously and cocompactly on a $\mathrm{CAT}(0)$ cube complex admits a projective Anosov representation into $\mathrm{SL}(d, \mathbb{R})$ for some $d$. More specifically, we show that if $\Gamma$ is a hyperbolic quasiconvex subgroup of a right-angled Coxeter group $C$, then a generic representation of $C$ by reflections restricts to a projective Anosov representation of $\Gamma$.