Fundamental groups of aspherical manifolds and maps of non-zero degree

TL;DR

This paper introduces a new class of groups called not IIPP and shows that certain aspherical manifolds with these groups do not admit non-zero degree maps from direct products, extending previous geometric and topological results.

Contribution

It defines the not IIPP class of groups, proves that aspherical manifolds with these groups lack non-zero degree maps from direct products, and characterizes product geometries in 4-manifolds.

Findings

Certain aspherical manifolds with not IIPP groups do not admit maps of non-zero degree from direct products.

Aspherical manifolds with reducible fundamental groups always admit such maps.

The property IIPP characterizes reducibility and implies vanishing simplicial volume.

Abstract

We define a new class of irreducible groups, called groups not infinite-index presentable by products or not IIPP. We prove that certain aspherical manifolds with fundamental groups not IIPP do not admit maps of non-zero degree from direct products. This extends previous results of Kotschick and Loeh, providing new classes of aspherical manifolds - beyond those non-positively curved ones which were predicted by Gromov - that do not admit maps of non-zero degree from direct products. A sample application is that an aspherical geometric 4-manifold admits a map of non-zero degree from a direct product if and only if it is a virtual product itself. This completes a characterization of the product geometries due to Hillman. Along the way we prove that for certain groups the property IIPP is a criterion for reducibility. This especially implies the vanishing of the simplicial volume of the corresponding aspherical manifolds. It is shown that aspherical manifolds with reducible fundamental groups do always admit maps of non-zero degree from direct products.