Infinite Volume and Infinite Injectivity Radius

TL;DR

This paper proves a conjecture of Margulis regarding the existence of arbitrarily large injected balls in certain geometric spaces associated with higher rank simple Lie groups, extending results to non-lattice subgroups.

Contribution

It establishes the existence of arbitrarily large injectivity radius in spaces from infinite covolume subgroups and extends rigidity results to stationary random subgroups in higher rank groups.

Findings

Locally symmetric spaces from infinite covolume subgroups have arbitrarily large injectivity radius.

Stationary random subgroups in higher rank groups exhibit stiffness and rigidity.

Stationary limits of measures on discrete subgroups are almost surely discrete.

Abstract

We prove the following conjecture of Margulis. Let $G$ be a higher rank simple Lie group and let $\Lambda\le G$ be a discrete subgroup of infinite covolume. Then, the locally symmetric space $\Lambda\backslash G/K$ 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 St\"{u}ck-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.