Invariant random subgroups (IRS) are conjugacy invariant probability measures on the space of subgroups of a given locally compact 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 can be reinterpreted by saying that for any probability measure preserving action of such a group, stabilizers are almost surely either trivial or "very big".

This generalizes an analogous result of Matsuzaki-Yabuki-Jaerisch for normal subgroups. 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.