Global smooth and topological rigidity of hyperbolic lattice actions

• 成果类型: Article
• 署名作者: Brown, Aaron; Hertz, Federico Rodriguez; Wang, Zhiren
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2017.186.3.3
• 发表日期: 2017
• 页码: 913-972
• 关键词: zariski-dense subgroups; semisimple lie-groups; local rigidity; anosov actions; invariant-measures; infinitesimal rigidity; cartan actions; tori; AUTOMORPHISMS; superrigidity

摘要

In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose Gamma is a lattice in a semisimple Lie group, all of whose factors have rank 2 or higher. Let a be alpha smooth Gamma-action on a compact nilmanifold M that lifts to an action on the universal cover. If the linear data rho of alpha contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of alpha and rho on a finite-index subgroup of Gamma. If alpha is a C-infinity action and contains an Anosov element, then the semiconjugacy is a C-infinity conjugacy. As a corollary, we obtain C-infinity global rigidity for Anosov actions by co-compact lattices in semisimple Lie groups 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.