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Word Measures on Unitary Groups

We combine concepts from random matrix theory and free probability together with ideas from the theory of commutator length in groups and maps from surfaces, and establish new connections between the two. More particularly, we study measures induced by free words on the unitary groups $U(n)$. Every word $w$ in the free group $F_r$ on $r$ generators determines a word map from $U(n)^r$ to $U(n)$, defined by substitutions. The $w$-measure on $U(n)$ is defined as the pushforward via this word map of the Haar measure on $U(n)^r$. Let $Tr_w(n)$ denote the expected trace of a random unitary matrix sampled from $U(n)$ according to the $w$-measure. It was shown by Voiculescu [Voic 91'] that for $w \ne 1$ this expected trace is $o(n)$ asymptotically in $n$. We relate the numbers $Tr_w(n)$ to the theory of commutator length of words and obtain a much stronger statement: $Tr_w(n)=O(n^{1-2g})$, where $g$ is the commutator length of $w$. Moreover, we analyze the number $\lim_{n\to\infty}n^{2g-1} \cdot Tr_w(n)$ and show it is an integer which, roughly, counts the number of (equivalence classes of) solutions to the equation $[u_1,v_1]...[u_g,v_g]=w$ with $u_i,v_i \in F_r$. Similar results are obtained for finite sets of words and their commutator length, and we deduce that one can 'hear' the stable commutator length of a word by 'listening' to its unitary measures.