Proper affine actions for right-angled Coxeter groups

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 in particular surface groups, hyperbolic $3$-manifold groups, and examples of word hyperbolic groups of arbitrarily large virtual cohomological dimension. We also study some examples in cohomological dimensions two and four, for which the dimension of the affine space may be substantially reduced.