Katok–Spatzier conjecture on algebraic models for higher-rank Anosov actions

Let $\mathbb{R}^k \times \mathbb{Z}^\ell \curvearrowright X$ be a transitive, $C^\infty$, Anosov action on a compact manifold without $C^\infty$ rank one factors. An algebraic action is an action on a homogeneous space $M \backslash G / \Gamma$ constructed from a Lie group $G$, a compact subgroup $M \subset G$, a cocompact lattice $\Gamma \subset G$, and a homomorphism $i: \mathbb{R}^k \times \mathbb{Z}^\ell \to \operatorname{Aff}_{M,\Gamma}(G)$, where $\operatorname{Aff}_{M,\Gamma}(G)$ consists of the affine maps preserving the relevant homogeneous-space data.

Katok–Spatzier conjecture. Up to finite cover, the action is $C^\infty$ conjugate to an algebraic action.

The conjecture seeks to identify transitive higher-rank Anosov actions without rank one factors with algebraic models. Such models include Weyl chamber flows and actions by toral automorphisms and are closed under products, suspensions, and skew products; the general classification remains open.