Abstract: 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, normally 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, Woess was able to show that when \Gamma is hyperbolic, \partial_R \Gamma coincides with 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.

*This talk is based on joint work with Matthieu Dussaule and Ilya Gekhtman.