Affine Anosov representations and proper actions

TL;DR

This paper introduces affine Anosov representations for hyperbolic groups into affine orthogonal groups, linking their properties to linear Anosov representations and proper group actions on Euclidean space.

Contribution

It defines affine Anosov representations and establishes their equivalence to linear Anosov representations with proper affine actions.

Findings

Affine Anosov representations are characterized by their linear parts being Anosov.

Such representations correspond to proper actions on Euclidean space.

The paper provides a new framework connecting hyperbolic groups, Anosov representations, and affine geometry.

Abstract

We define the notion of affine Anosov representations of word hyperbolic groups into the affine group $\mathsf{SO}^0(n+1,n)\ltimes\mathbb{R}^{2n+1}$. We then show that a representation $\rho$ of a word hyperbolic group is affine Anosov if and only if its linear part $\mathtt{L}_\rho$ is Anosov in $\mathsf{SO}^0(n+1,n)$ with respect to the stabilizer of a maximal isotropic plane and $\rho(\Gamma)$ acts properly on $\mathbb{R}^{2n+1}$.