Joshua Frisch (California Institute of Technology)

Title: The ICC property in random walks and dynamics.

Abstract: A topological dynamical system (i.e. a group acting by homeomorphisms on a compact Hausdorff space) is said to be proximal if for any two points $p$ and $q$ we can simultaneously "push them together" (rigorously, there is a net $g_n$ such that $\lim g_n(p) = \lim g_n(q)$). In his paper introducing the concept of proximality, Glasner noted that whenever $\mathbb{Z}$ acts proximally, that action will have a fixed point. He termed groups with this fixed point property “strongly amenable”. The Poisson boundary of a random walk on a group is a measure space that corresponds to the space of different asymptotic trajectories that the random walk might take. Given a group $G$ and a probability measure $\mu$ on $G$, the Poisson boundary is trivial (i.e. has no non-trivial events) if and only if $G$ supports a bounded $\mu$-harmonic function. A group is called Choquet–Deny if its Poisson boundary is trivial for every $\mu$. In this talk I will discuss work giving an explicit classification of which groups are Choquet–Deny, which groups are strongly amenable, and what these mysteriously equivalent classes of groups have to do with the ICC property. I will also discuss why strongly amenable groups can be viewed as strengthening amenability in at least three distinct ways, thus proving the name is extremely well deserved.

Link: https://mcgill.zoom.us/j/98910726246?pwd=VHlzTzdTZGtqcHVuWGNKdys4d0FzQT09

Zoom ID: 989 1072 6246
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