For any word \(w\) in a free group of rank \(r>0\), and any compact group \(G\), the word \(w\) induces a ‘word map’ from \(G^r\to G\) defined by substitutions. We may also choose the \(r\) elements of \(G\) independently with respect to Haar measure on \(G\), and then apply the word map. This gives us a random element of \(G\) whose distribution depends on \(w\). An interesting observation is that this distribution doesn't change if we change \(w\) by an automorphism of the free group.

This invites various questions. Which algebraic properties of \(w\) are reflected in the distribution it induces on compact groups? Do these distributions on compact groups determine \(w\) up to automorphisms? In general these questions are wide open.

My talk will be about the case \(G = U(n)\), the \(n\times n\) complex unitary matrices.

In the first half of my talk I'll prove some basic but interesting facts about word-induced measures. I'll give more precise versions of the questions stated above and explain what is known to date.

In the second half I'll discuss the technical tool we use to study these problems. It is a precise formula for the moments of the distribution induced by \(w\) on \(U(n)\), in terms of purely algebraic invariants of \(w\). In the formula, there is a surprising appearance of concepts from infinite group theory, more specifically, Euler characteristics of mapping class groups of surfaces.