Co-universality for Toeplitz algebras of random walks on relatively hyperbolic groups

When studying quotients of C-algebras generated by creation and annihilation operators on analogues of Fock space, the question of the existence of a co-universal quotient plays an important role in answering fundamental questions in the theory. The study of co-universal quotients goes back to works of Cuntz, and Cuntz and Krieger, on uniqueness theorems for C-algebras arising from symbolic dynamics, and by now co-universal quotients have been shown to exist in several broad classes of examples of Toeplitz C*-algebras.

When associating Toeplitz C-algebras to random walks on a group $G$, new notions of *ratio-limit space and boundary emerge from searching for their co-universal quotients, and the existence of these co-universal quotients becomes intimately related to the group dynamics on the ratio-limit boundary.

In this talk I will exlain how we extended results of Woess to show that there is co-universal quotient for a large class of symmetric random walks on relatively hyperbolic groups. This sheds light on some questions of Woess on ratio-limits for random walks on relatively hyperbolic groups, and extends a result mine on the existence of co-universal quotients for Toeplitz C*-algebras for random walks.

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