Stationary random subgroups and injectivity radius of hyperbolic manifolds

Ilya Gekhtman (Technion Haifa)

Jul 21. 2023, 15:10 — 15:50

There is a long tradition of using probabilistic methods to solve geometric problems. I will present one such result. Namely, I will show that  if the bottom of the spectrum of the Laplacian on a hyperbolic manifold (or more generally rank one locally symmetric space) M  is equal to that of its universal cover then M has points with arbitrary large injectivity radius. 


This is (in some sense the optimal) rank 1 analogue of a recent result of Fraczyk-Gelander which asserts that any infinite volume higher rank locally symmetric space has  points with arbitrary large injectivity radius. 


The proof will depend on a probabilistic result showing that non-free stationary actions of G have "large" stabilizers.  Namely, if the stabilizers are discrete then they have full limit sets and exponential growth rate greater than half of the entropy divided by the drift of the random walk, in particular bounded away from 0. Joint work with Arie Levit.

Further Information
ESI Schrödinger and Boltzmann Lecture Hall
Associated Event:
Geometric and Asymptotic Group Theory with Applications 2023 - Groups and Dynamics (Workshop)
Christopher Cashen (U of Vienna)
Javier de la Nuez González (KIAS, Seoul)
Alexandra Edletzberger (U of Vienna)
Yash Lodha (U of Hawaii, Honolulu)