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Self-similarity of the generalized Baumslag-Solitar groups

Dessislava H. Kochloukova

Abstract

We show that all residually finite generalized Baumslag-Solitar groups of rank $n \geq 1$, defined on a finite and connected graph, are self-similar. Furthermore we prove that all residually finite fundamental groups of (finite, connected) graph of groups where all vertex and edge groups are torsion-free and commensurable with the Heisenberg group and all edge groups properly embed in the corresponding vertex groups are self-similar.

Self-similarity of the generalized Baumslag-Solitar groups

Abstract

We show that all residually finite generalized Baumslag-Solitar groups of rank , defined on a finite and connected graph, are self-similar. Furthermore we prove that all residually finite fundamental groups of (finite, connected) graph of groups where all vertex and edge groups are torsion-free and commensurable with the Heisenberg group and all edge groups properly embed in the corresponding vertex groups are self-similar.
Paper Structure (7 sections, 16 theorems, 97 equations)

This paper contains 7 sections, 16 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fundamental groups of graph of groups and residual finiteness
  4. Self-similar groups and virtual endomorphisms
  5. Automata generating free products of groups of order 2
  6. Proofs
  7. Ascending HNN extension of the Heisenberg group is self-similar

Key Result

Theorem 1.1

Let $G$ be a generalized Baumslag-Solitar group of rank n. Then the following conditions are equivalent: i) $G$ is self-similar; ii) $G$ is residually finite; iii) $G$ is linear over $\mathbb{Q}$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • ...and 9 more