A criterion for Tits alternative on the centralizer of a matrix

We give a necessary and sufficient condition on a matrix for its centralizer in $\mathsf{G}\mathsf{L}(\mathsf{n},\mathbb{Z})$ 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 $\mathsf{G}\mathsf{L}(\mathsf{n},\mathbb{Z})$. 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 $\mathsf{G}\mathsf{L}(\mathsf{n},\mathbb{Z})$ by Eick-Hofmann-O’Brien and to an orbit problem for polycyclic groups, by Eick and Ostheimer.