The conjecture that Anosov manifolds are virtually nilmanifolds

Let $M$ be a compact, connected manifold, and let an Anosov diffeomorphism mean a diffeomorphism of $M$ with the hyperbolic tangent-space splitting described in the source. A nilmanifold is a homogeneous space $G/\Gamma$, where $G$ is a nilpotent Lie group and $\Gamma$ is a discrete subgroup of $G$.

The conjecture from the 1960s. If there is an Anosov diffeomorphism $f$ on $M$, then some finite cover of $M$ is a nilmanifold. Furthermore, lifting $f$ to the finite cover yields an affine map on the nilmanifold.

This is a classification problem for compact manifolds admitting Anosov diffeomorphisms. The conjecture asserts that, after passing to a finite cover, every such manifold has nilmanifold structure and the lifted dynamics are affine; the source presents it among fundamental questions that remain open.