Every word ω in a free group naturally induces a probability measure on every compact group G. For example, if ω = [x, y] is the commutator word, a random element sampled by the ω-measure is given by the commutator [g, h] of two independent, Haar-random elements of G. Back in 1896, Frobenius showed that if G is a finite group and ψ an irreducible character, then the expected value of ψ([g, h]) is \({1 \over {\psi \left(e \right)}}\). This is true for any compact group, and completely determines the [x, y]-measure on these groups. An analogous result holds with the commutator word replaced by any surface word.

We prove a converse to this theorem: if ω induces the same measure as [x, y] on every compact group, then, up to an automorphism of the free group, ω is equal to [x, y]. The same holds when [x, y] is replaced by any surface word.

The proof relies on the analysis of word measures on unitary groups and on orthogonal groups, which appears in separate papers, and on new analysis of word measures on generalized symmetric groups that we develop here.

一个自由组中的每个单词ω自然会在每个紧致组G上诱发一个概率测度。例如，如果ω = [ x，y ]是换向词，则由ω-测度采样的随机元素由换向器[ g ，h ]是G的两个独立的Haar随机元素。早在1896年，Frobenius表明，如果G是一个有限群而ψ是一个不可约性，则ψ（[ g，h ]）的期望值为\（{ 1 \ over {\ psi \ left（e \ right）}} \）。这对于任何紧致组都是正确的，并且完全确定[ x，y对这些群体采取措施。类似的结果保持不变，换向器字被任何表面字所代替。

我们证明了与该定理相反的情况：如果ω在每个紧致群上诱导出与[ x，y ]相同的量度，那么直到自由群的自同构性为止，ω等于[ x，y ]。将[ x，y ]替换为任何表面词时，也是如此。

证明依赖于单独论文中出现的unit类和正交组上的字测度分析，以及我们在此处开发的针对广义对称群上的字测度的新分析。