Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Isomorphism classification of Leary-Minasyan groups

Motiejus Valiunas

TL;DR

The paper provides a complete isomorphism classification for Leary–Minasyan groups $G(A,L)$, a family of $\mathbb{Z}^n$-based HNN-extensions, by splitting into non-metabelian, non-polycyclic metabelian, and polycyclic regimes. It develops tools from HNN-extension theory, ascending HNN-extensions, and Bass–Serre tree actions to derive explicit invariants: A and L data, conjugacy relations in $GL_n(\mathbb{Z})$ and $GL_n(\mathbb{Q})$, and finite-index subgroup behavior. The main theorem states that two such groups are isomorphic if and only if their defining data satisfy one of three corresponding conditions, which recover known BS-classifications as special cases and provide a comprehensive framework for this family. The results leverage geometric group theory techniques like tree actions and rigidity theorems, linking algebraic invariants to the geometry of the Bass–Serre trees. Overall, the work gives a precise, case-split criterion for when $G(A,L)\cong G(\overline{A},\overline{L})$, clarifying how polycyclic, metabelian, and non-metabelian structures influence isomorphism types.

Abstract

Recently, I. J. Leary and A. Minasyan studied the class of groups $G(A,L)$ defined as commensurating HNN-extensions of $\mathbb{Z}^n$. This class, containing the class of Baumslag-Solitar groups, also includes other groups with curious properties, such as being CAT(0) but not biautomatic. In this paper, we classify the groups $G(A,L)$ up to isomorphism.

Isomorphism classification of Leary-Minasyan groups

TL;DR

The paper provides a complete isomorphism classification for Leary–Minasyan groups , a family of -based HNN-extensions, by splitting into non-metabelian, non-polycyclic metabelian, and polycyclic regimes. It develops tools from HNN-extension theory, ascending HNN-extensions, and Bass–Serre tree actions to derive explicit invariants: A and L data, conjugacy relations in and , and finite-index subgroup behavior. The main theorem states that two such groups are isomorphic if and only if their defining data satisfy one of three corresponding conditions, which recover known BS-classifications as special cases and provide a comprehensive framework for this family. The results leverage geometric group theory techniques like tree actions and rigidity theorems, linking algebraic invariants to the geometry of the Bass–Serre trees. Overall, the work gives a precise, case-split criterion for when , clarifying how polycyclic, metabelian, and non-metabelian structures influence isomorphism types.

Abstract

Recently, I. J. Leary and A. Minasyan studied the class of groups defined as commensurating HNN-extensions of . This class, containing the class of Baumslag-Solitar groups, also includes other groups with curious properties, such as being CAT(0) but not biautomatic. In this paper, we classify the groups up to isomorphism.

Paper Structure

This paper contains 12 sections, 20 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Prelimilaries and division into cases
  3. HNN-extensions and centralisers
  4. Ascending HNN-extensions and semidirect products
  5. Actions on trees
  6. Coarse classification of Leary--Minasyan groups
  7. Non-metabelian groups
  8. Non-polycyclic metabelian groups
  9. Polycyclic groups
  10. Isomorphisms
  11. Conjugacy over Q
  12. The proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $n,\overline{n} \geq 0$, let $A \in GL_n(\mathbb{Q})$, $\overline{A} \in GL_{\overline{n}}(\mathbb{Q})$, and let $L \leq \mathbb{Z}^n \cap A^{-1}(\mathbb{Z}^n)$, $\overline{L} \leq \mathbb{Z}^{\overline{n}} \cap \overline{A}^{-1}(\mathbb{Z}^{\overline{n}})$ be finite index subgroups. Then $G(A,L

Theorems & Definitions (38)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Remark 1.4
  • Proposition 2.1: Britton's Lemma, see serre
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • Proposition 2.4: see culler
  • Lemma 2.5
  • ...and 28 more