Word measures on groups

Let G be a finite group. Every word w in the free group on, say, 2 generators x and y, induces a probability measure on G as follows: sample two uniformly random elements g and h from G and evaluate w(g,h) to get an element of G. For example, if w = xyxy^{-2}, one obtains the random element ghgh^{-2} in G. The study of word measures on groups has revealed a lot of beautiful structure with surprising connections to other fields of mathematics, and has shed light on the finite group G as well as on the free group. 

In this talk I will survey some of what we know and some of the challenges ahead, and focus on one illustrating example.

This is based on joint works with Liam Hanany, Michael Magee, Chen Meiri, Ori Parzanchevski and Danielle West.