Matrix Group Integrals, Surfaces, and Mapping Class Groups I: $U(n)$

Since the 1970's, physicists and mathematicians who study random matrices in the GUE or GOE models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces. We establish a new aspect of this theory: for random matrices sampled from the group $\mathcal{U}\left(n\right)$ of unitary matrices. More concretely, we study measures induced by free words on $\mathcal{U}\left(n\right)$. Let $F_{r}$ be the free group on $r$ generators. To sample a random element from $\mathcal{U}\left(n\right)$ according to the measure induced by $w\in F_{r}$, one substitutes the $r$ letters in $w$ by $r$ independent, Haar-random elements from $\mathcal{U}\left(n\right)$. The main theme of this paper is that every moment of this measure is determined by families of pairs $\left(\Sigma,f\right)$, where $\Sigma$ is an orientable surface with boundary, and $f$ is a map from $\Sigma$ to the bouquet of $r$ circles, which sends the boundary components of $\Sigma$ to powers of $w$. A crucial role is then played by Euler characteristics of subgroups of the mapping class group of $\Sigma$. As corollaries, we obtain asymptotic bounds on the moments, we show that the measure on $\mathcal{U}\left(n\right)$ bears information about the number of solutions to the equation $\left[u_{1},v_{1}\right]\cdots\left[u_{g},v_{g}\right]=w$ in the free group, and deduce that one can ``hear'' the stable commutator length of a word through its unitary word measures.