Affine Anosov Representations and Proper Actions

Abstract

We define the notion of affine Anosov representations of word hyperbolic groups into the affine group $\textsf {SO}^0(n+1,n)\ltimes {\mathbb {R}}^{2n+1}$. 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 $\textsf {SO}^0(n+1,n)$ with respect to the stabilizer of a maximal isotropic plane and $\rho (\Gamma )$ acts properly on $\mathbb {R}^{2n+1}$.

Schottky presentations of positive representations

“…This result was extended by Ghosh-Treib in [GT17] for representations with Anosov linear part, and it is their result that we will use here to show properness of our actions. Though not the most general restatement of the result of that paper, the formulation most convenient for us is: Proposition 4.6 ([GT17]).…”

“…We reprove properness via a different method not relying on the machinery of diffused Margulis invariants from [GT17]. We construct fundamental domains and show that they tile all of affine space: Theorem 1.2.…”

Entropy and affine actions for surface groups

“…Now by [10, Theorem 7.1 and Definition 4.4] if there is a measure that annihilates the Labourie–Margulis diffusion (defined in equation (10)), then the action on the affine group is not proper. This concludes the proof of Danciger and Zhang's Theorem 1.1.…”

“…On the other hand, Margulis work in [21] has exhibited free groups acting properly on the affine space. Work of Goldman, Margulis and the author [11], further extended by Ghosh and Treib [10], has shown how to characterize proper actions of a hyperbolic group using the Labourie–Margulis diffusion , which is an extension to measures — introduced in [17] — of the Margulis invariant introduced by Margulis in [22]. As for surface groups, they were shown by Mess [23] to admit no proper affine actions on the affine 3‐space.…”

“…The Labourie–Margulis diffusion $$M$$ is a continuous function on the space of finite measures invariant by the geodesic flow of the surface, associated to the representation on the affine space [17]. According to a generalization of [11, 17] due to Ghosh and Treib [10, Theorem 7.1 and Definition 4.4], if there exists a measure $$\mu$$ so that $$M(\mu )=0$$, then the action on the affine space is not proper.…”