Stationary Actions of Countable Groups and What Are They Good For

Invariant random subgroups are conjugation invariant measures on the space of subgroups of a given group G. Their study provides a fertile playground for the interaction of algebra, geometry, probability and ergodic theory.

On one hand, they arise as point stabilizers for probability measure preserving actions. 

On the other hand, they provide a vast generalization of both normal subgroups of countable groups and lattices in Lie groups.

On the third hand, they (and their relatives, the so-called stationary random subgroups) have applications ranging from proving Margulis's celebrated normal subgroup theorem (which asserts that any normal subgroup of a lattice in a higher rank simple Lie group is finite index) to compactifying the moduli space of Riemann surfaces, to studying the injectivity radius of hyperbolic manifolds. In this talk, I will give an overview of invariant and stationary random subgroups and their applications. Special attention will be given to invariant random subgroups of semisimple Lie groups and of hyperbolic groups.