Logic Seminar

A topological dynamical system (i.e. a group acting by homeomorphisms on a compact topological space) is said to be proximal if for any two points p and q we can simultaneously push them together i.e. there is a sequence $g_n$ such that $lim g_n(p)=lim g_n (q)$. In his paper introducing the concept of proximality Glasner noted that whenever $\Z$ acts proximally that action will have a fixed point. He termed groups with this fixed point property "strongly amenable" and showed that non-amenable groups are not strongly amenable and virtually nilpotent groups are strongly amenable. In this talk I will discuss recent work precisely characterizing which (countable) groups are strongly amenable. In particular I will show a group is strongly amenable if and only if it has no infinite conjugacy class (ICC) groups as factors. This, as a corollary, proves that the only finite generated strongly amenable groups are virtually nilpotent. The proof technique is to show that, for a special class of symbolical systems, a generic (i.e. comeagre) action is proximal.

This is joint work with Omer Tamuz and Pooya Vahidi Ferdowski.