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

TL;DR

This paper investigates the properties of complete affine manifolds with partially hyperbolic holonomy groups, establishing bounds on their cohomological dimensions based on hyperbolic index and using coarse geometry techniques.

Contribution

It provides new bounds on the cohomological dimension of affine groups with partially hyperbolic holonomy, linking hyperbolic index to geometric and algebraic properties.

Findings

If the holonomy group is partially hyperbolic of index k< n/2, then cd(Γ) ≤ n-k.

For finitely-presented affine groups with k-Anosov linear part, cd(Γ) ≤ n-k.

Existence of a compact collection of affine subspaces where the group acts.

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.