Conjugacy separability of groups of integer matrices

Abstract:

An element g of a group G is conjugacy distinguished if and only if given any element h of G either g is conjugate to h or there is a homomorphism $\xi$ of G onto a finite group such that $\xi (g)$ is not conjugate to $\xi (h)$. Following A. W. Mostowski, a group is conjugacy separable if every one of its elements is conjugacy distinguished. Let ${\text {GL}}(n,Z)$ be the group of $n \times n$ integer matrices with determinant $\pm 1$. Let ${\text {SL}}(n,Z)$ be the subgroup of ${\text {GL}}(n,Z)$ consisting of matrices with determinant $+ 1$. It is shown that ${\text {GL}}(n,Z)$ and ${\text {SL}}(n,Z)$ are conjugacy separable if and only if $n = 1$ or 2. The groups ${\text {SL}}(n,Z)$ are also called unimodular groups. Let ${\text {GL}}(n,{Z_p})$ be the group of invertible p-adic integer matrices and ${\text {SL}}(n,{Z_p})$ be the group of p-adic integer matrices with determinant 1. It is shown that ${\text {GL}}(n,{Z_p})$ and ${\text {SL}}(n,{Z_p})$ are conjugacy separable for all n and all p.