Commensurators of abelian subgroups of biautomatic groups

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 and Minasyan (Commensurating HNN-extensions: non-positive curvature and biautomaticity, Geom Topol 25:1819–1860, 2021) are either biautomatic or cannot arise as subgroups of biautomatic groups, answering a question posed in Leary and Minasyan (Commensurating HNN-extensions: non-positive curvature and biautomaticity, Geom Topol 25:1819–1860, 2021) 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.