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Infinite Volume and Infinite Injectivity Radius

Mikolaj Fraczyk, Tsachik Gelander

Abstract

We prove the following conjecture of Margulis. Let $G$ be a higher rank simple Lie group and let $Λ\le G$ be a discrete subgroup of infinite covolume. Then, the locally symmetric space $Λ\backslash G/K$ admits injected balls of any radius. This can be considered as a geometric interpretation of the celebrated 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 Stück-Zimmer theorem 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.

Infinite Volume and Infinite Injectivity Radius

Abstract

We prove the following conjecture of Margulis. Let be a higher rank simple Lie group and let be a discrete subgroup of infinite covolume. Then, the locally symmetric space admits injected balls of any radius. This can be considered as a geometric interpretation of the celebrated 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 Stück-Zimmer theorem 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.

Paper Structure

This paper contains 18 sections, 54 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. The statement for simple groups
  3. Confined subgroups of semisimple groups
  4. Random walk on $G/\Lambda$
  5. Stiffness and the Stuck--Zimmer theorem for stationary measures
  6. The conclusion
  7. Random walks on the space of discrete subgroups
  8. The measure associated to $G$
  9. The discreteness radius
  10. The Margulis function on the space of discrete subgroups of $G$
  11. Stationary limit are discrete
  12. Essential results about discrete stationary random subgroups
  13. A decomposition result for Invariant Random Subgroups
  14. Stationary measures of rank one groups
  15. Stiffness of discrete stationary random subgroups for higher rank groups
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $G$ be a connected centre-free simple Lie group of real rank at least $2$ and let $X=G/K$ be the associated symmetric space. Let $\Lambda\le G$ be a discrete group of infinite covolume. Then for every $r>0$ there is a point $p\in \Lambda\backslash X$ where the injectivity radius is at least $r$.

Theorems & Definitions (103)

  • Theorem 1.1
  • Example 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 93 more