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Commensurators of abelian subgroups of biautomatic groups

Motiejus Valiunas

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.

Commensurators of abelian subgroups of biautomatic groups

Abstract

We show that the commensurator of any finitely generated abelian subgroup in a biautomatic group centralises a finite-index subgroup of . 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.

Paper Structure

This paper contains 9 sections, 15 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Commensurators
  4. Biautomatic groups
  5. Quasiconvex subsets and subgroups
  6. Polyhedral functions
  7. Geometry of polyhedral functions
  8. A polyhedral function associated to a biautomatic structure on Z^n
  9. The proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $A \in GL_n(\mathbb{Q})$, and let $L$ be a finite-index subgroup of $\mathbb{Z}^n \cap A^{-1}(\mathbb{Z}^n)$. Then $G(A,L)$ is a subgroup of a biautomatic group if and only if $A$ has finite order. In particular, there exist CAT(0) groups that are not embeddable into biautomatic groups.

Figures (1)

  • Figure 1: A representation of a polyhedral function $f\colon \mathbb{R}^2 \to \mathbb{R}$. See Examples \ref{['ex:intro']}, \ref{['ex:section']} and \ref{['ex:biauto']} for details.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3: see epstein
  • Definition 2.4
  • Theorem 2.5: see gersten-short
  • Lemma 2.6
  • proof
  • ...and 22 more