Ratio-limit boundaries for random walks on relatively hyperbolic groups.

Adam Dor-On (Haifa U.)

Jul 18. 2023, 14:50 — 15:30

The ratio-limit boundary \partial_R \Gamma for a random walk on a group \Gamma is defined as the remainder obtained after compactifying \Gamma with respect to ratio-limit kernels H(x,y) = \lim_n \frac{P^n(x,y)}{P^n(e,y)}, where P^n(x,y) is the probability to pass from x to y in n steps. These limits were shown to exist for various classes of groups, usually by establishing a local limit theorem which measures the asymptotic behavior of P^n(x,y) as n \rightarrow \infty. In increasing level of generality, works of Woess, Lalley, Gouezel and Dussaule establish such local limit theorems for certain symmetric random walks on relatively hyperbolic groups. Techniques developed in these works then allow us to study \partial_R \Gamma. For instance, when \Gamma is hyperbolic, Woess was able to show that \partial_R \Gamma is the Gromov boundary of \Gamma

In this talk we will explain how for a large class of random walks on relatively hyperbolic groups, the ratio-limit boundary is essentially minimal. That is, there is a unique minimal closed \Gamma-invariant subspace of \partial_R \Gamma. This result is motivated by applications in operator algebras, and indeed, by using it we are able to show the existence of a co-universal equivariant quotient of Toeplitz C*-algebras for a large class of random walks on relatively hyperbolic groups. The talk will focus mostly on geometry, topology, and dynamics, and if time permits I will explain some operator algebraic aspects and motivation.

*Based on joint work with Matthieu Dussaule and Ilya Gekhtman.

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)