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Affine Anosov representations and proper actions

Sourav Ghosh, Nicolaus Treib

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 $ρ$ of a word hyperbolic group is affine Anosov if and only if its linear part $\mathtt{L}_ρ$ is Anosov in $\mathsf{SO}^0(n+1,n)$ with respect to the stabilizer of a maximal isotropic plane and $ρ(Γ)$ acts properly on $\mathbb{R}^{2n+1}$.

Affine Anosov representations and proper actions

Abstract

We define the notion of affine Anosov representations of word hyperbolic groups into the affine group . We then show that a representation of a word hyperbolic group is affine Anosov if and only if its linear part is Anosov in with respect to the stabilizer of a maximal isotropic plane and acts properly on .

Paper Structure

This paper contains 10 sections, 19 theorems, 104 equations.

Table of Contents

  1. Consistent orientations
  2. Gromov geodesic flow
  3. Anosov representations
  4. Affine Anosov representations
  5. Affine deformations of Anosov representations
  6. Margulis spacetimes
  7. Margulis spacetimes vs Affine Anosov representations
  8. Appendix
  9. (AMS)-Proximality
  10. Sections over the flow space

Key Result

Theorem 1

A representation of a word hyperbolic group $\Gamma$ into $\mathsf{SO}^0(n+1,n)\ltimes\mathbb{R}^{2n+1}$ is affine Anosov if and only if its linear part is Anosov with respect to the stabilizer of a maximal isotropic plane and it acts properly on $\mathbb{R}^{2n+1}$.

Theorems & Definitions (40)

  • Theorem 1
  • Lemma 1.1
  • proof
  • Proposition 1.2
  • Proposition 1.3
  • Proposition 2.1: BK
  • Proposition 2.2: Mineyev
  • Lemma 2.3
  • proof
  • Definition 3.1
  • ...and 30 more