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∈GLm(C) be quasi-unipotent matrices such that the Jordan Canonical Form of B consists of a single block, and suppose that for all k⩾0 the matrix ABk is also quasi-unipotent. Then A and B have a common eigenvector. In particular, 〈A,B〉<GLm(C) is a solvable subgroup. We give applications of this result to the representation theory of mapping class groups of orientable surfaces.