Surface words are determined by word measures on groups

Every word $w$ in a free group naturally induces probability measure on every compact $G$. For example, if $w=\left[x,y\right]$ is the commutator word, random element sampled by $w$-measure given $\left[g,h\right]$ of two independent, Haar-random elements Back 1896, Frobenius showed that $G$ finite and $\psi$ an irreducible character, then expected value $\psi\left(\left[g,h\right]\right)$ $\frac{1}{\psi\left(e\right)}$. This true for any group, completely determines $\left[x,y\right]$-measure these groups. An analogous result holds with replaced surface word. We prove converse to this theorem: same as $\left[x,y\right]$ then, up automorphism equal $\left[x,y\right]$. The when proof relies analysis measures unitary groups orthogonal groups, which appears separate papers, new generalized symmetric we develop here.