A Bass--Serre theoretic proof of a theorem of Burns and Romanovskii

TL;DR

This paper offers a new proof of a theorem by Burns and Romanovskii, demonstrating that free products of subgroup separable groups are also subgroup separable, using Bass's graph of groups theory.

Contribution

It introduces a Bass--Serre theoretic proof of the subgroup separability of free products, providing a novel perspective compared to previous proofs.

Findings

Confirmed subgroup separability is preserved under free products

Applied Bass's graph of groups theory to subgroup separability proofs

Provided a new, more geometric proof approach

Abstract

A well known theorem of Burns and Romanovskii states that a free product of subgroup separable groups is itself subgroup separable. We provide a proof using the language of immersions and coverings of graphs of groups, due to Bass.