Proper affine actions for right-angled Coxeter groups

TL;DR

This paper constructs proper affine actions for right-angled Coxeter groups on certain orthogonal groups and Lie algebras, extending to virtually special groups including surface and hyperbolic 3-manifold groups, with dimension reductions in specific cases.

Contribution

It provides explicit constructions of proper affine actions for right-angled Coxeter groups and virtually special groups, expanding understanding of their geometric representations.

Findings

Proper affine actions constructed for right-angled Coxeter groups.

Virtually special groups admit proper affine actions on some Euclidean space.

Dimension reduction achieved in specific cohomological cases.

Abstract

For any right-angled Coxeter group $\Gamma$ on $k$ generators, we construct proper actions of $\Gamma$ on $\mathrm{O}(p,q+1)$ by right and left multiplication, and on the Lie algebra $\mathfrak{o}(p,q+1)$ by affine transformations, for some $p,q\in\mathbb N$ with $p+q+1=k$. As a consequence, any virtually special group admits proper affine actions on some $\mathbb R^n$: this includes e.g. surface groups, hyperbolic 3-manifold groups, examples of word hyperbolic groups of arbitrarily large virtual cohomological dimension, etc. We also study some examples in cohomological dimension two and four, for which the dimension of the affine space may be substantially reduced.