Vanishing simplicial volume for certain affine manifolds

TL;DR

This paper proves that certain affine manifolds with specific holonomy properties have zero simplicial volume, supporting the Auslander Conjecture and providing a cohomological criterion for vanishing simplicial volume in aspherical manifolds.

Contribution

It establishes a new vanishing result for simplicial volume in affine manifolds with particular holonomy conditions and offers a cohomological criterion for such vanishing.

Findings

Affine manifolds with injective holonomy containing pure translations have zero simplicial volume.

Provides a cohomological criterion for vanishing simplicial volume in aspherical manifolds with normal amenable subgroups.

Answers a special case of a question posed by Lück.

Abstract

We show that closed aspherical manifolds supporting an affine structure, whose holonomy map is injective and contains a pure translation, must have vanishing simplicial volume. This provides some further evidence for the veracity of the Auslander Conjecture. Along the way, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups of $\pi_1$ to have vanishing simplicial volume. This answers a special case of a question due to L\"uck.