Virtual retraction properties in groups

TL;DR

This paper investigates the properties of virtual retraction in groups, focusing on properties (LR) and (VRC), their stability under various group constructions, and characterizes virtual free factors in virtually free groups.

Contribution

It introduces new criteria for virtual free factors, analyzes the stability of (LR) and (VRC) under group operations, and settles a conjecture on virtually free groups.

Findings

(VRC) is stable under finite index overgroups.

(LR) is not stable under finite index overgroups.

A simple criterion for virtual free factors in virtually free groups.

Abstract

If $G$ is a group, a virtual retract of $G$ is a subgroup which is a retract of a finite index subgroup. Most of the paper focuses on two group properties: property (LR), that all finitely generated subgroups are virtual retracts, and property (VRC), that all cyclic subgroups are virtual retracts. We study the permanence of these properties under commensurability, amalgams over retracts, graph products and wreath products. In particular, we show that (VRC) is stable under passing to finite index overgroups, while (LR) is not. The question whether all finitely generated virtually free groups satisfy (LR) motivates the remaining part of the paper, studying virtual free factors of such groups. We give a simple criterion characterizing when a finitely generated subgroup of a virtually free group is a free factor of a finite index subgroup. We apply this criterion to settle a conjecture of Brunner and Burns.