This is joint work with Michael Magee. 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 $U(n)$. For example, if $w$ is a word in $F_2 = \langle x,y \rangle$, sample at random two elements from $U(n)$, $A$ for $x$ and $B$ for $y$, and evaluate $w(A,B)$. The measure of this random element is called the $w$-measure on $U(n)$. We study the expected trace (and other invariants) of a random unitary matrix sampled from $U(n)$ according to the $w$-measure, and find surprising algebraic properties of $w$ that determine these quantities.