If uniformly random substitutions from a permutation group \(S_n\) in a word \(w\) always result in the uniform distribution, then \(w\) must be a primitive word. It is not known if the same holds when the permutation groups are replaced by the unitary matrix groups \(U(n)\). This problem, mentioned in the concluding section, motivates the study of distribution of words in compact groups, which is the underlying topic of the current paper.
For words \(w_1,\dots,w_\ell\) in the free group, consider the expected value \(T_{w_1,\dots,w_\ell}(n)\) of the product \(\operatorname{tr}(w_1)\cdots \operatorname{tr}(w_\ell)\), where the \(w_i\) are evaluated at matrices chosen uniformly at random from the unitary group \(U(n)\). It is shown that the value of \(T_{w_1,\dots,w_\ell}\) is a rational function of (large enough) \(n\), with rational coefficients.
The main theme of this paper is that the coefficients of the Laruent series of \(T_{w_1,\dots,w_\ell}(n)\) are determined by combinatorial data related to the \(w_j\). Namely, one considers pairs \((\Sigma,f)\) where \(\Sigma\) is an oriented, compact surface with \(\ell\) boundary components and no closed connected components; and \(f\) is a map from \(\Sigma\) to a bouquet of \(\ell\) loops, such that the restriction of \(f\) to the boundary represents the words \(w_j\). Applying a mixture of combinatorial geometry techniques, the authors prove that the coefficient of \(n^{-c}\) in the Laurent series of \(T_{w_1,\dots,w_n}\) is the sum of \(L^2\)-Euler characteristics of the stabilizers of \(f\) in the mapping class group of \(\Sigma\), ranging over equivalence classes of pairs \((\Sigma,f)\) with \(\chi(\Sigma) = -c\).
An interesting application is that the leading coefficient of \(T_w(n)\) counts the presentations of \(w\) as a product of minimal length of commutators, up to equivalence, “correcting” for nontrivial stabilizers. Similarly, the stable commutator length can be read off from the orders of magnitude of the mixed values \(T_{w^{j_1},\dots,w^{j_p}}(n)\).
The main object in the proof is a polysimplicial complex of transverse maps homotopic to \(f\), associated to a pair \((\Sigma,f)\). This complex is shown to be contractible. One is then able to obtain a combinatorial formula for the above-mentioned \(L^2\)-Euler characteristic.