Aspherical Products Which do not Support Anosov Diffeomorphisms

Abstract

Abstract. We show that the product of infranilmanifolds with certain aspherical closed manifolds do not support Anosov diffeomorphisms. As a special case, we obtain that products of a nilmanifold and negatively curved manifolds of dimension at least 3 do not support Anosov diffeomorphisms.

Anosov diffeomorphisms of products I. Negative curvature and rational homology spheres

“…In Section 2.1, we will see how this class can be used to rule out transitive Anosov diffeomorphisms on negatively curved manifolds, which is Yano's Theorem 1.1 (d). We note that in dimensions higher than two the transitivity assumption in Yano's result is in fact not needed as explained in [7,Cor. 4.5]; we will discuss this briefly in Section 2.2.…”

“…Outer automorphism groups. As explained 1 by Gogolev-Lafont [7], aspherical manifolds whose fundamental group has torsion outer automorphism group do not support transitive Anosov diffeomorphisms. For let M be an aspherical manifold with torsion Out(π 1 (M)) and suppose that there exists an Anosov diffeomorphism f : M −→ M. Since Out(π 1 (M)) is torsion, there is some l such that (π 1 (f )) l is an inner automorphism of π 1 (M), and so ((π 1 (f )) l ) * : H * (π 1 (M); R) −→ H * (π 1 (M); R) is the identity.…”

“…Moreover, if n is odd, then M × S n does not support Anosov diffeomorphisms. (b) (Gogolev-Lafont [7]). Let N be a closed infranilmanifold and M be a closed smooth aspherical manifold such that π 1 (M) is Hopfian, the outer automorphism group Out(π 1 (M)) is finite, and the intersection of all maximal nilpotent subgroups of π 1 (M) is trivial.…”

Anosov diffeomorphisms of products II. Aspherical manifolds

“…Theorem 3.1 ( [25,7]). If M is a negatively curved manifold, then M does not support Anosov diffeomorphisms.…”

“…when g = h = 1) admits Anosov diffeomorphisms. However, the case of Σ g × Σ h , where at least one of g or h is ≥ 2, seems to be more subtle: [7,Section 7.2]). Does the product of two closed aspherical surfaces at least one of which is hyperbolic admit an Anosov diffeomorphism?…”