An effective Lie--Kolchin theorem for quasi-unipotent matrices

We establish an effective version of the classical Lie–Kolchin Theorem. Namely, let $A,B\in {\mathrm{G}\mathrm{L}}_m(\mathbb{C})$ be quasi–unipotent matrices such that the Jordan Canonical Form of $B$ consists of a single block, and suppose that for all $k\ge 0$ the matrix $A{B}^k$ is also quasi–unipotent. Then $A$ and $B$ have a common eigenvector. In particular, $\langle A,B\rangle<$\mathrm{G}\mathrm{L}$_m($\mathbb{C}$)$ is a solvable subgroup. We give applications of this result to the representation theory of mapping class groups of orientable surfaces.