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

TL;DR

This paper presents an effective version of the Lie--Kolchin Theorem for quasi-unipotent matrices, showing under certain conditions that such matrices share a common eigenvector, with applications to mapping class group representations.

Contribution

It provides a new effective criterion for quasi-unipotent matrices to have a common eigenvector, extending classical results with applications in geometric group theory.

Findings

Quasi-unipotent matrices with a single Jordan block share a common eigenvector.

The subgroup generated by such matrices is solvable.

Applications to the representation theory of mapping class groups.

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 $k\geq0$ 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.