## Geometry and Topology

•  Ilya Gekhtman , University of Toronto
•  Growth rates of invariant random subgroups of hyperbolic groups and rank 1 Lie groups.
•  02/21/2019
•  2:00 PM - 3:00 PM
•  C304 Wells Hall

Abstract: Invariant random subgroups (IRS) are conjugacy invariant probability measures on the space of subgroups of a given group G. They arise naturally as point stabilizers of probability measure preserving actions. The space of invariant random subgroups of SL_{2}R can be regarded as a natural compactification of the moduli space of Riemann surfaces, related to the Deligne-Mumford compactification. Invariant random subgroups can be regarded as a generalization both of normal subgroups and of lattices in topological groups. 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 Lie group, or a hyperbolic group) is almost surely greater than half the Hausdorff dimension of the boundary. This generalizes an analogous result of Matsuzaki-Yabuki-Jaerisch for normal s As a corollary, we obtain that if $\Gamma$ is a typical subgroup and $X$ a rank 1 symmetric space then $\lambda_{0}(X/\Gamma)<\lambda_{0}(X)$ where $\lambda_0$ is the bottom of the spectrum of the Laplacian. 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.

## Contact

Department of Mathematics
Michigan State University