Global smooth and topological rigidity of hyperbolic lattice actions

Abstract

In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices.Suppose Γ is a lattice in semisimple Lie group, all of whose factors have rank 2 or higher. Let α be a smooth Γ-action on a compact nilmanifold M that lifts to an action on the universal cover. If the linear data ρ of α contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of α and ρ, on a finite-index subgroup of Γ. If α is a C ∞ action and contains an Anosov element, then the semiconjugacy is a C ∞ conjugacy.As a corollary, we obtain C ∞ global rigidity for Anosov actions by cocompact lattices in semisimple Lie group with all factors rank 2 or higher. We also obtain global rigidity of Anosov actions of SL(n, Z) on T n for n ≥ 5 and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds. Contents

On the work of Rodriguez Hertz on rigidity in dynamics

“…Very recently, A. Brown, F. Rodriguez Hertz and Z. Wang found a major improvement [BHW15]. We will discuss it below in Section 6.3.…”

The Structure of Homeomorphism and Diffeomorphism Groups

“…Theorem 0.9 (Brown-Rodriguez-Hertz-Wang [BRHW17]). Suppose Γ is an irreducible cocompact lattice in a semisimple Lie group acting on a nilmanifold by smooth diffeomorphisms.…”

Book Review: Group actions in ergodic theory, geometry, and topology: Selected papers

“…The Brown-Fisher-Hurtado argument uses cocycle superrigidity in accord with Zimmer's basic template but makes an important shift of perspective toward the problem of uniformizing estimates of subexponential derivative growth across invariant measures, a reorientation inspired by advances in the rigidity theory of Anosov Z d -actions. To produce invariant measures, the proof elaborates on the "nonresonance implies invariance" principle that had recently been developed by Brown, Rodriguez Hertz, and Wang [4], making use of ideas and results from measure rigidity that concern the algebraic nature of invariant measures in higher rank situations, including Ratner's theorems on unipotent flows and the deployment of Ledrappier and Young's work on entropy. The smooth metric whose existence leads to the conclusion (via a standard appeal to Margulis superrigidity) is obtained by a novel application of Lafforgue's strong property (T).…”