Groups and Dynamics Seminar

Date: April 28, 2021

Time: 12:00PM - 1:00PM

Location: online

Speaker: Andy Zucker, UCSD

Title: Topological Furstenberg boundaries of C*-simple groups.

Abstract: Given a group G, a boundary action of G is an action of G on a compact space which is minimal and strongly proximal. The topological Furstenberg boundary of a group G, denoted \Pi_s(G), is the largest boundary action it admits. It is non-trivial if and only if G is non-amenable. More recently, Kalantar and Kennedy have shown that a countable discrete group G is C*-simple if and only if G acts freely on \Pi_s(G). However, in the setting of abstract topological dynamics, one can ask much more precise questions about the size of \Pi_s(G), even knowing that it is free. At one extreme, perhaps there is a C*-simple group G where \Pi_s(G) is "almost metrizable," i.e. a highly proximal extension of some metrizable boundary action. For other C*-simple groups, perhaps one can find a continuum-sized family of "mutually disjoint" free boundary actions, which in some sense means that \Pi_s(G) is as large as possible. This talk will discuss the open question of whether every C*-simple group falls into the latter category; while the talk will have more questions than answers, we will discuss joint work in progress with Tianyi Zheng giving an affirmative answer for free groups.