Supersolvable closures of finitely generated subgroups of the free group

TL;DR

This paper proves that the pro-supersolvable closure of any finitely generated subgroup of a free group is itself finitely generated, extending known results for pro-p and pro-nilpotent closures.

Contribution

It introduces the concept of pro-supersolvable closures and establishes their finite generation, expanding the understanding of subgroup closures in free groups.

Findings

Pro-supersolvable closure of finitely generated subgroups is finitely generated.

Extends results from pro-p and pro-nilpotent closures to supersolvable closures.

Provides new insights into subgroup closure properties in free groups.

Abstract

We prove the pro-supersolvable closure of a finitely generated subgroup of the free group is finitely generated. It extends similar results for pro-$p$ closures proved by Ribes-Zalesskii and pro-Nilpotent closures proved by Margolis-Sapir-Weil.