  We establish an effective version of the classical Lie--Kolchin Theorem.
Namely, let $A,B\in\mathrm{GL}_m(\mathbb{C})$ be quasi--unipotent matrices such
that the Jordan Canonical Form of $B$ consists of a single block, and suppose
that for all $k\geq0$ the matrix $AB^k$ is also quasi--unipotent. Then $A$ and
$B$ have a common eigenvector. In particular, $\langle
A,B\rangle<\mathrm{GL}_m(\mathbb{C})$ is a solvable subgroup. We give
applications of this result to the representation theory of mapping class
groups of orientable surfaces.
