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

Abstract

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 $k\ge 0$ the matrix $A{B}^k$ is also quasi--unipotent. Then $A$ and $B$ have a common eigenvector. In particular, $⟨A,B⟩<{\mathrm{GL}}_m(\mathbb{C})$ is a solvable subgroup. We give applications of this result to the representation theory of mapping class groups of orientable surfaces.

19 pages