Convex cocompactness for Coxeter groups

TL;DR

This paper characterizes which Coxeter groups can be represented as convex cocompact reflection groups in projective space, linking hyperbolic and nonhyperbolic cases and providing new examples in the theory of convex cocompact subgroups.

Contribution

It fully describes the spaces of convex cocompact Coxeter group representations via Cartan matrices, including hyperbolic and nonhyperbolic cases, and connects to projective Anosov representations.

Findings

All infinite, word hyperbolic Coxeter groups admit convex cocompact reflection representations.

Such hyperbolic groups correspond exactly to projective Anosov representations.

The paper provides numerous examples of nonhyperbolic Coxeter groups with convex cocompact representations.

Abstract

We investigate representations of Coxeter groups into $\mathrm{GL}(n,\mathbb{R})$ as geometric reflection groups which are convex cocompact in the projective space $\mathbb{P}(\mathbb{R}^n)$. We characterize which Coxeter groups admit such representations, and we fully describe the corresponding spaces of convex cocompact representations as reflection groups, in terms of the associated Cartan matrices. The Coxeter groups that appear include all infinite, word hyperbolic Coxeter groups; for such groups the representations as reflection groups that we describe are exactly the projective Anosov ones. We also obtain a large class of nonhyperbolic Coxeter groups, thus providing many examples for the theory of nonhyperbolic convex cocompact subgroups in $\mathbb{P}(\mathbb{R}^n)$ developed in arXiv:1704.08711.