Colloquium: Tuesday, November 15, 2022. Speaker: Ilya Gekhtman (Technion). Title: “Random walks and the geometry of discrete subgroups of rank 1 Lie groups”.

Our next Math colloquium talk will be on Tuesday, 15th of November, in room 614, Science & Education building.

Speaker: Ilya Gekhtman (Technion – IIT)

Date: Tuesday, November 15th, 2022

Time: 14:00

Title: Random walks and the geometry of discrete subgroups of rank 1 Lie groups.


Invariant random subgroups (IRS) are conjugation invariant probability measures on the space of subgroups in a given group G. They arise as point stabilizers of probability measure preserving actions. In a sense that I will describe, invariant random subgroups can be regarded as a generalization both of normal subgroups and of lattices in Lie groups. As such, it is interesting to extend results from the theories of normal subgroups and of lattices to the IRS setting. A more general notion is a stationary random subgroup (SRS) where the measure on the space of subgroups is no longer required to be conjugation invariant, but only stationary with respect to some random walk. SRS are useful in studying IRS which are in themselves useful for studying lattices.

For higher rank Lie groups it turns out IRS and SRS are extremely rigid: they all arise from conjugating a single lattice. For rank 1 Lie groups the space of IRS is so big as to be unmanageable but we can still find order in this chaos.

To this end, 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. We prove a related bound for SRS, with “half” replaced by entropy divided by drift of the random walk. All terms except “group” and “measure” will be define in the course of the talk.