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Realizing residually finite groups as subgroups of branch groups

Steffen Kionke, Eduard Schesler

Abstract

We prove that every finitely generated, residually finite group $G$ embeds into a finitely generated perfect branch group $Γ$ such that many properties of $G$ are preserved under this embedding. Among those are the properties of being torsion, being amenable, and not containing a non-abelian free group. As an application we construct a finitely generated, non-amenable torsion branch group.

Realizing residually finite groups as subgroups of branch groups

Abstract

We prove that every finitely generated, residually finite group embeds into a finitely generated perfect branch group such that many properties of are preserved under this embedding. Among those are the properties of being torsion, being amenable, and not containing a non-abelian free group. As an application we construct a finitely generated, non-amenable torsion branch group.
Paper Structure (5 sections, 13 theorems, 18 equations)

This paper contains 5 sections, 13 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Groups acting on rooted trees
  3. Branch groups
  4. Stabilized sections and shrinking sequences
  5. Proof of the main theorem

Key Result

Theorem 1.1

Every finitely generated, residually finite group $G$ embeds into a finitely generated perfect branch group $\Gamma$ such that

Theorems & Definitions (28)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Corollary 1.3
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • ...and 18 more