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Property (LR) and an embedding theorem for virtually free groups

Ashot Minasyan

Abstract

We prove that every virtually free group $G$ has property (LR) of Long and Reid: each finitely generated subgroup of $G$ is a retract of a finite index subgroup. The main ingredient in the proof is a new embedding result stating that every countable virtually free group embeds in a double of a finite group. As a corollary, we show that any group commensurable with the direct product of a free group and a finitely generated abelian group has (LR). This applies to generalized Baumslag-Solitar groups of arbitrary rank $n \in \mathbb{N}$ with finite monodromy, which, in particular, include all non-cyclic one-relator groups with center.

Property (LR) and an embedding theorem for virtually free groups

Abstract

We prove that every virtually free group has property (LR) of Long and Reid: each finitely generated subgroup of is a retract of a finite index subgroup. The main ingredient in the proof is a new embedding result stating that every countable virtually free group embeds in a double of a finite group. As a corollary, we show that any group commensurable with the direct product of a free group and a finitely generated abelian group has (LR). This applies to generalized Baumslag-Solitar groups of arbitrary rank with finite monodromy, which, in particular, include all non-cyclic one-relator groups with center.
Paper Structure (6 sections, 20 theorems, 29 equations)

This paper contains 6 sections, 20 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Embedding virtually free groups
  3. Embedding general virtually free groups in multiple special HNN-extensions
  4. Embedding finitely generated virtually free groups in doubles of finite groups
  5. Property (LR) for virtually free groups
  6. More groups with (LR)

Key Result

Theorem 1.1

Virtually free groups have property (LR).

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Conjecture 1.5
  • Conjecture 1.6
  • Theorem 2.1: KPS and DD
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 25 more