Isomorphism classification of Leary-Minasyan groups

Recently, I. J. Leary and A. Minasyan studied the class of groups $G(A,L)$ defined as commensurating HNN-extensions of ${\mathbb{Z}}^n$. This class, containing the class of Baumslag-Solitar groups, also includes other groups with curious properties, such as being CAT(0) but not biautomatic. In this paper, we classify the groups $G(A,L)$ up to isomorphism.