Cartan actions of higher rank abelian groups and their classification

Spatzier, Ralf; Vinhage, Kurt

doi:10.1090/jams/1033

Cartan actions of higher rank abelian groups and their classification
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by Ralf Spatzier and Kurt Vinhage

J. Amer. Math. Soc. 37 (2024), 731-859

DOI: https://doi.org/10.1090/jams/1033

Published electronically: August 31, 2023

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Abstract:

We study $\mathbb {R}^k \times \mathbb {Z}^\ell$ actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program.

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