Subgroup separability in residually free groups

TL;DR

This paper proves that finitely presentable subgroups of residually free groups are separable and that certain subgroups are virtual retracts, providing a uniform solution to the membership problem in this context.

Contribution

It establishes subgroup separability and virtual retraction properties for finitely presentable subgroups of residually free groups, advancing understanding of their subgroup structure.

Findings

Finitely presentable subgroups are separable.

Subgroups of type FP_infinity are virtual retracts.

A uniform solution to the membership problem is provided.

Abstract

We prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type $\mathrm{FP}_\infty$ are virtual retracts. We describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups.