Arithmeticity, superrigidity, and totally geodesic submanifolds

• 成果类型: Article
• 署名作者: Bader, Uri; Fisher, David; Miller, Nicholas; Stover, Matthew
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2021.193.3.4
• 发表日期: 2021
• 页码: 837-861
• 关键词: commensurability classes; SUBGROUPS

摘要

Let Gamma be a lattice in SO0(n, 1). We prove that if the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least 2, then Gamma is arithmetic. This answers a question of Reid for hyperbolic n-manifolds and, independently, McMullen for hyperbolic 3-manifolds. We prove these results by proving a superrigidity theorem for certain representations of such lattices. The proof of our superrigidity theorem uses results on equidistribution from homogeneous dynamics, and our main result also admits a formulation in that language.