Cartan Actions of Higher Rank Abelian Groups and their Classification

Ralf Spatzier; Kurt Vinhage

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arXiv · 1901.06559

Cartan Actions of Higher Rank Abelian Groups and their Classification

Ralf Spatzier·Kurt Vinhage

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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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Ralf Spatzier, Kurt Vinhage. 2019-01-19. Cartan Actions of Higher Rank Abelian Groups and their Classification. https://arxiv.org/abs/1901.06559

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