Commensurators of abelian subgroups of biautomatic groups

TL;DR

This paper proves that in biautomatic groups, the commensurator of any finitely generated abelian subgroup centralizes a finite-index subgroup, and it classifies certain CAT(0) groups as either biautomatic or not, providing new examples.

Contribution

It establishes a centralization property for commensurators of abelian subgroups in biautomatic groups and classifies specific CAT(0) groups in relation to biautomaticity, answering open questions.

Findings

Commensurator of abelian subgroup centralizes a finite-index subgroup.

Certain CAT(0) groups are either biautomatic or not subgroups of biautomatic groups.

First examples of CAT(0) groups not embeddable in biautomatic groups.

Abstract

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.