Topological volumes of certain complete affine manifolds
Alberto Casali, Marco Moraschini
TL;DR
The paper proves that closed connected complete affine manifolds whose fundamental group contains an infinite amenable normal subgroup satisfy cat_Am(M) ≤ dim(M), leading to vanishing of the simplicial volume, stable integral simplicial volume, and minimal volume entropy. The authors combine a topological approach via injective Seifert fiber spaces with Kamishima’s torus-fiber structure result to obtain the bound, and then apply Gromov-type vanishing theorems to deduce the volume vanishings. These results partially answer Lück's question in the complete affine case and yield new instances where integral approximation holds. The work also provides an algebraic criterion connecting virtually poly-Z kernels with aspherical manifolds of zero volume and discusses consequences and open problems in the broader setting of aspherical manifolds.
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 Lück 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.