Vadim Kaimanovich, Université d'Ottawa

Title: Arboreal structures on groups and the associated boundaries - I

Abstract: Geometers are very much used to endowing manifolds with additional structures (Riemannian, symplectic, etc.). Likewise, a group can be equipped with a probability measure which gives rise to the associated random walk (in the same way as a Riemannian metric gives rise to the associated Brownian motion). This setup produces the Poisson boundary as a measure space which describes the stochastically significant behaviour of the sample paths of the random walk at infinity.

The Poisson boundary can usually be identified with the natural geometric boundary of the group (provided the latter is well-defined) endowed with an appropriate hitting distribution (e.g., for hyperbolic groups, cf. the classical Poisson formula for the hyperbolic plane in the disk model). Still, a number of very natural questions about the identification of the Poisson boundary (or even just about its non-triviality) for general groups remain wide open. It has been known for nearly 40 years that amenable groups can be characterized as the ones for which there *exists* a measure with the trivial Poisson boundary. According to a very recent result of Frisch - Hartman - Tamuz - Vahidi Ferdowsi the hyper-FC-central groups are precisely those for which the boundary of *any* measure is trivial. The key ingredient of this result is the existence of a measure with a non-trivial boundary on any group with infinite conjugacy classes (ICC). I will talk about a joint work with Anna Erschler, in which we introduce a new arboreal structure on an arbitrary ICC group and subsequently identify the Poisson boundary in terms of this structure.