  We show that the commensurator of any finitely generated abelian subgroup $H$
in a biautomatic group centralises a finite-index subgroup of $H$. We deduce
that the CAT(0) groups introduced by Leary-Minasyan are either biautomatic or
cannot arise as subgroups of biautomatic groups, answering a question posed by
Leary-Minasyan and generalising an analogous result for Baumslag-Solitar
groups. These are the first examples of CAT(0) groups that are not subgroups of
biautomatic groups.

Group Theory