Complete affine manifolds with Anosov holonomy groups II: partially hyperbolic holonomy and cohomological dimensions

Abstract

Let $N$ be a complete affine manifold $A^n/\Gamma$ of dimension $n$ where $\Gamma$ is an affine transformation group and $K(\Gamma, 1)$ is realized as a finite CW-complex. $N$ has a partially hyperbolic holonomy group if the tangent bundle pulled over the unit tangent bundle over a sufficiently large compact part splits into expanding, neutral, and contracting subbundles along the geodesic flow. We show that if the holonomy group is partially hyperbolic of index $k$, $k < n/2$, then $\mathrm{cd}(\Gamma) \leq n-k$. Moreover, if a finitely-presented affine group $\Gamma$ acts on $A^n$ properly discontinuously and freely with the $k$-Anosov linear group 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 on. The techniques here are mostly from coarse geometry.