Let $n$ be an integer. Andreas Thom mentioned that Camille Jordan showed that there exists some $m \in {\mathbb N}$ (depending on $n$), such that for any pair of $n \times n$-unitaries $u,v \in U(n)$ which generate a finite group, one has $[u^m,v^m] = 1_n$. In particular, there are two distinct words of length $\ell=2m $ over the alphabet $\{u,v\} $ that represent the same group element.

My question is

what is known about how small such an $m_n$ or $\ell_n $ can be?

The connection with the Jordan-Schur Theorem is that if $G$ is a finite group generated by $u$ and $v$, and $H$ a normal abelian subgroup of index $m$, then $u^mH=H$ and $v^mH=H$ (since any group element raised to the order of the group is the identity) and so $u^m$ and $v^m$ belong to $H$, hence, $H$ being abelian, they commute. So if $[G:H]=m$ with $H$ abelian then we have such an $m$.

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