Anosov diffeomorphisms of products II. Aspherical manifolds

Abstract

A long-standing conjecture asserts that any Anosov diffeomorphism of a closed manifold is finitely covered by a diffeomorphism which is topologically conjugate to a hyperbolic automorphism of a nilpotent manifold. In this paper, we show that any closed 4-manifold that carries a Thurston geometry and is not finitely covered by a product of two aspherical surfaces does not support (transitive) Anosov diffeomorphisms.

Endomorphisms of mapping tori

“…As explained already, the free factors F r i do not satisfy the conditions of Theorem 1.3, and so the same is true for K. Let θ : K K be an automorphism. By [Ne1], since F r i and π 1 (Σ g j ) are Hopfian and have trivial center, there exists an integer m such that…”