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

TL;DR

This paper proves that virtually free groups possess property (LR), meaning each finitely generated subgroup is a retract of a finite index subgroup, using a new embedding theorem for countable virtually free groups.

Contribution

The paper introduces a novel embedding theorem for countable virtually free groups and demonstrates that these groups have property (LR), extending to groups commensurable with certain product groups.

Findings

Virtually free groups have property (LR).

Countable virtually free groups embed in doubles of finite groups.

Groups commensurable with free and abelian groups also have (LR).

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.