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

02.11.2023, 10:15  –  Campus Golm, Haus 14, Room 0.21
Forschungsseminar: Gruppen und Operatoralgebren

Adam Dor-On (Haifa University)

When studying quotients of C*-algebras generated by creation and annihilation operators on analogues of Fock space, the existence of a unique smallest equivariant quotient becomes an important question in the theory. When it exists, this quotient is sometimes called the co-universal quotient. The study goes back to uniqueness theorems of Cuntz, and Cuntz and Krieger, which were extended by many authors to include several broad classes of examples.

When associating Toeplitz C*-algebras to random walks, new notions of ratio-limit space and 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.