A criterion for Tits alternative on the centralizer of a matrix

TL;DR

This paper establishes a criterion for when the centralizer of a matrix in GL(n,Z) is polycyclic or abelian, providing an effective approach to the conjugacy problem in related arithmetic groups.

Contribution

It introduces a necessary and sufficient condition for centralizers in GL(n,Z) to be polycyclic or abelian, enabling effective solutions to the conjugacy problem in certain arithmetic groups.

Findings

Characterization of matrices with polycyclic centralizers

Effective criterion for abelian centralizers

Solution to the conjugacy problem in arithmetic groups

Abstract

We give a necessary and sufficient condition on a matrix for its centralizer in $\sf{GL}(n,\mathbb{Z})$ to be polycyclic, or equivalently in this case, not to contain a non-abelian free subgroup. We give a simple condition on the matrix ensuring that it is abelian. This can be thought of as an effective Tits alternative on centralizers in $\sf{GL}(n,\mathbb{Z})$. We apply these criteria to the conjugacy problem in certain arithmetic groups preserving a non-degenerate $\mathbb{Q}$-bilinear form, such as integral symplectic groups. We derive an effective solution to the conjugacy problem in such groups when given matrices satisfy the above criterion. This solution is based on effective solutions to the conjugacy problem in $\sf{GL}(n,\mathbb{Z})$ by Eick-Hofmann-O'Brien and to an orbit problem for polycyclic groups, by Eick and Ostheimer.