The ratio-limit boundary for a random walk on a group is defined as the remainder obtained after compactifying with respect to ratio-limit kernels , where is the probability to pass from to in 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 as . 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 . For instance, when is hyperbolic, Woess was able to show that is the Gromov boundary of .
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 -invariant subspace of . 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.