Infinite volume and infinite injectivity radius

• 成果类型: Article
• 署名作者: Fraczyk, Mikolaj; Gelander, Tsachik
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2023.197.1.6
• 发表日期: 2023
• 页码: 389-421
• 关键词: SUBGROUPS; stabilizers; lattices

摘要

We prove the following conjecture of Margulis. Let G be a higher rank simple Lie group, and let ? <= G be a discrete subgroup of infinite covol-ume. Then, the locally symmetric space ?\G/K admits injected balls of any radius. This can be considered as a geometric interpretation of the cel-ebrated Margulis normal subgroup theorem. However, it applies to general discrete subgroups not necessarily associated to lattices. Yet, the result is new even for subgroups of infinite index of lattices. We establish similar results for higher rank semisimple groups with Kazhdan's property (T). We prove a stiffness result for discrete stationary random subgroups in higher rank semisimple groups and a stationary variant of the Stuck-Zimmer the-orem for higher rank semisimple groups with property (T). We also show that a stationary limit of a measure supported on discrete subgroups is almost surely discrete.