Topological volumes of certain complete affine manifolds

TL;DR

This paper estimates the amenable category and simplicial volume of certain complete affine manifolds, showing they have zero simplicial volume and related invariants, thus answering a question by Lück.

Contribution

It provides the first estimate of the amenable category for these manifolds and proves their simplicial volume vanishes, addressing an open question.

Findings

All such manifolds have zero simplicial volume.

They satisfy integral approximation.

Stable integral simplicial volume and minimal volume entropy also vanish.

Abstract

We provide an estimate of the amenable category of oriented closed connected complete affine manifolds whose fundamental group contains an infinite amenable normal subgroup. As an application we show that all such manifolds have zero simplicial volume. This answers a question by L\"uck in the case of complete affine manifolds. Our construction also provides the vanishing of stable integral simplicial volume and minimal volume entropy. This means that such manifolds satisfy integral approximation.