Ratio-limits and Toeplitz quotients for random walks on relatively hyperbolic groups.

Adam Dor-On (Haifa University)

Thursday 2nd March, 2023 16:00-17:00 Maths 311B


When studying quotients of C*-algebras generated by creation and annihilation operators on analogues of Fock space, the existence of a unique \emph{smallest equivariant quotient} becomes an important question in the theory. When it exists, this quotient is sometimes called the \emph{co-universal quotient}. The study goes back to works of Cuntz, and Cuntz and Krieger, which was extended by many authors, on \emph{uniqueness theorems} for C*-algebras arising from directed graphs, and has been gradually extended to include several broad classes of examples.

When associating Toeplitz C*-algebras to random walks, new notions of \emph{ratio-limit space} and \emph{ratio-limit boundary} emerge from computing natural quotients, and the question of co-universality becomes intimately related to the geometry and the dynamics on the boundary of the random walk.

In this talk I will explain how we extended results of Woess and myself to show that there is a co-universal quotient for a large class of symmetric random walks on relatively hyperbolic groups. This allowed us to make significant progress on some questions of Woess on ratio-limits for random walks on relatively hyperbolic groups, and shed light on the general question of co-universality for Toeplitz C*-algebras arising from subproduct systems.

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

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