Franks–Manning conjecture on Anosov diffeomorphisms

Let $M$ be a closed manifold and let $f\colon M\to M$ be an Anosov diffeomorphism, meaning a diffeomorphism with a hyperbolic splitting of the tangent bundle. A hyperbolic infra-nilmanifold automorphism is an affine automorphism of an infra-nilmanifold whose linear part has no eigenvalue of absolute value $1$. Franks–Manning conjecture. The map $f$ is topologically conjugate to a hyperbolic infra-nilmanifold automorphism.

This conjecture asserts that hyperbolic infra-nilmanifold automorphisms provide essentially the only examples of Anosov diffeomorphisms. Its general status is open.