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 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.

arXiv comment41 pages, 4 figures

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Cited by in corpus (5)

• Anosov representations with Lipschitz limit set
• Proper affine actions: a sufficient criterion
• Margulis Multiverse: Infinitesimal Rigidity, Pressure Form and Convexity
• Special cubulation of strict hyperbolization
• Representations having vectors fixed by a Levi subgroup