On Livsic's Theorem, Superrigidity, and Anosov Actions of Semisimple Lie Groups

TL;DR

This paper generalizes Livsic's Theorem to broader dynamical systems, strengthening Zimmer's results on algebraic hulls of Anosov actions and establishing a Holder geometric structure for certain actions.

Contribution

It extends Livsic's Theorem and combines it with superrigidity to analyze Anosov actions of semisimple Lie groups, revealing new geometric structures.

Findings

Generalized Livsic's Theorem for specific dynamical systems

Strengthened Zimmer's results on algebraic hulls

Established Holder geometric structures for multiplicity free Anosov actions

Abstract

We prove a generalization of Livsic's Theorem on the vanishing of the cohomology of certain types of dynamical systems. As a consequence, we strengthen a result due to Zimmer concerning algebraic hulls of Anosov actions of semisimple Lie groups. Combining this with Topological Superrigidity, we find a Holder geometric structure for multiplicity free Anosov actions.