Complete affine manifolds with Anosov holonomy groups

TL;DR

This paper establishes a deep connection between affine manifolds with Anosov holonomy groups and their geometric and algebraic properties, using representation theory and coarse geometry techniques.

Contribution

It proves the equivalence between P-Anosov holonomy groups and partially hyperbolic holonomy groups for affine manifolds, a novel result over the full general linear group.

Findings

Complete affine manifolds with P-Anosov holonomy are characterized by partial hyperbolicity.

Bounds on the cohomological dimension of the fundamental group based on hyperbolicity index.

Existence of invariant affine subspaces where the group acts properly discontinuously.

Abstract

Let $N$ be a complete affine manifold $\mathbb{A}^n/\Gamma$ of dimension $n$, where $\Gamma$ is an affine transformation group acting on the complete affine space $\mathbb{A}^n$, and $K(\Gamma, 1)$ is realized as a finite CW-complex. $N$ has a $k$-partially hyperbolic holonomy group if the tangent bundle pulled back over the unit tangent bundle of a sufficiently large compact subset splits into expanding, neutral, and contracting subbundles along the geodesic flow, where the expanding and contracting subbundles are $k$-dimensional with $k < n/2$. In part 1, we will demonstrate that the complete affine $n$-manifold has a $P$-Anosov linear holonomy group for a parabolic subgroup $P$ of $\mathrm{GL}(n, \mathbb{R})$ if and only if it has a partially hyperbolic linear holonomy group. This had never been done over the full general linear group before this paper. Part 1 will primarily employ representation theory techniques. In part 2, we demonstrate that if the holonomy group is partially hyperbolic of index $k$, where $k < n/2$, then $\mathrm{cd}(\Gamma) \leq n-k$. Moreover, if a finitely-presented affine group $\Gamma$ acts properly discontinuously and freely on $\mathbb{A}^n$ with a $k$-Anosov linear subgroup for $k \leq n/2$, then $\mathrm{cd}(\Gamma) \leq n-k$. Also, there exists a compact collection of $n-k$-dimensional affine subspaces where $\Gamma$ acts. The techniques employed here mostly stem from the coarse geometry theory. Canary and Tsouvalis previously proved the same result using the powerful method of Bestvina and Mess for word hyperbolic groups; however, our approach differs in that our method projects the holonomy cover to a stable affine subspace, and we plan to generalize to relative Anosov groups.