Regularity of conjugacies of algebraic actions of Zariski dense groups

TL;DR

This paper proves that under certain conditions, topological conjugacies between algebraic group actions and nearby smooth actions are actually smooth, especially for Zariski dense subgroups acting on homogeneous spaces.

Contribution

It establishes smoothness of conjugacies for algebraic actions of Zariski dense groups on homogeneous spaces, extending rigidity results to broader classes of group actions.

Findings

Topological conjugacies are smooth under specified conditions.

Results apply to Zariski dense subgroups of SL_d(Z) on tori.

Applicable to actions of simple Lie groups on homogeneous spaces.

Abstract

Let \alpha_0 be an affine action of a discrete group \Gamma on a compact homogeneous space X and \alpha_1 a smooth action of \Gamma on X which is C^1-close to \alpha_0. We show that under some conditions, every topological conjugacy between \alpha_0 and \alpha_1 is smooth. In particular, our results apply to Zariski dense subgroups of SL_d(Z) acting on the torus T^d and Zariski dense subgroups of a simple noncompact Lie group G acting on a compact homogeneous space X of G with an invariant measure.