Infinite volume and infinite injectivity radius

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 \rm (T). We prove a stiffness result for discrete stationary random subgroups in higher rank semisimple groups and a stationary variant of the Stuck–Zimmer theorem for higher rank semisimple groups with property \rm (T). We also show that a stationary limit of a measure supported on discrete subgroups is almost surely discrete.