## Dynamics, Geometry, & Groups Seminar

### Friday, September 13th, 2019

**Time:** 10:30 a.m** Place:** Jeffery Hall 319

**Speaker:** James Mingo (Queen's University)

**Title:** Can a free group have a fractional number of generators? In the free world it can.

**Abstract:** Starting with a countable discrete group G we complete the group ring in a suitable topology to get L(G), the group von Neumann algebra. We let L(n) be the group von Neumann algebra for the free group on n ≥ 2 generators. Using random matrix theory Voiculescu showed that if we tensor L(m) with the k x k matrices we get an algebra isomorphic to L(n). The relation between k, m, and n is the same as in Schreier's index theorem for subgroups of free groups. With this we can define L(t) for any real t > 1 as the group algebra of the free group with t generators. I will explain the main ideas in the proof. No prior knowledge of free probability will be assumed.