Complete affine manifolds with Anosov holonomy groups

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.