Ilya Gekhtman, University of Toronto

Growth rates of invariant random subgroups in negative curvature

Invariant random subgroups (IRS) are conjugacy invariant probability measures on the space of subgroups in a given group G. They arise as point stabilizers of probability measure preserving actions. Invariant random subgroups can be regarded as a generalization both of normal subgroups and of lattices. As such, it is interesting to extend results from the theories of normal subgroups and of lattices to the IRS setting. Jointly with Arie Levit, we prove such a result: the critical exponent (exponential growth rate) of an infinite IRS in an isometry group of a Gromov hyperbolic space (such as a rank 1 symmetric space, or a hyperbolic group) is almost surely greater than half the Hausdorff dimension of the boundary. If the subgroup is of divergence type, we show its critical exponent is in fact equal to the dimension of the boundary. If G has property (T) we obtain as a corollary that an IRS of divergence type must in fact be a lattice. The proof uses ergodic theorems for actions of hyperbolic groups. I will also talk about results about growth rates of normal subgroups of hyperbolic groups that inspired this work.