For any right-angled Coxeter group on generators, we construct proper actions of on by right-and-left multiplication and on the Lie algebra by affine transformations, for some with . As a consequence, any virtually special group admits proper affine actions on some : this includes in particular surface groups, hyperbolic -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.
Jeffrey Danciger. François Guéritaud. Fanny Kassel. "Proper affine actions for right-angled Coxeter groups." Duke Math. J. 169 (12) 2231 - 2280, 1 September 2020. https://doi.org/10.1215/00127094-2019-0084