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

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.