  Commensurating HNN-extensions of $\mathbb{Z}^n$, or Leary-Minasyan groups,
have recently been studied by I. J. Leary and A. Minasyan. These groups are
higher-dimensional analogues of Baumslag-Solitar groups. Among other results,
Leary-Minasyan groups were shown to serve as first known examples of CAT(0)
groups that are not biautomatic.
  It is well-known that a Baumslag-Solitar groups are either biautomatic or not
embeddable in biautomatic groups. Motivated by this, we may ask whether a
similar dichotomy holds for Leary-Minasyan groups. In this paper, we answer
this question in the affirmative, thus providing examples of CAT(0) groups that
are not subgroups of biautomatic groups. The proof relies on studying
commensurators of arbitrary finitely generated abelian subgroups in biautomatic
groups.
