Anosov diffeomorphisms of products II. Aspherical manifolds

TL;DR

This paper investigates conditions under which certain aspherical manifolds, specifically products involving infranilmanifolds and others with specific fundamental group properties, do not support Anosov diffeomorphisms, extending previous results.

Contribution

It weakens existing conditions to show that products of infranilmanifolds with specific aspherical manifolds lack Anosov diffeomorphisms and proves a new stability result for the Hopf property in group theory.

Findings

Products of infranilmanifolds with certain aspherical manifolds do not support Anosov diffeomorphisms.

The product of finitely many Hopfian groups with trivial center is Hopfian.

Extension of conditions for non-existence of Anosov diffeomorphisms on complex manifolds.

Abstract

We study aspherical manifolds that do not support Anosov diffeomorphisms. Weakening conditions of Gogolev and Lafont, we show that the product of an infranilmanifold with finitely many aspherical manifolds whose fundamental groups have trivial center and finite outer automorphism group does not support Anosov diffeomorphisms. In the course of our study, we obtain a result of independent group theoretic and topological interest on the stability of the Hopf property, namely, that the product of finitely many Hopfian groups with trivial center is Hopfian.