An Uncountable Family of Finitely Generated Residually Finite Groups

TL;DR

This paper constructs an uncountable family of finitely generated residually finite groups by doubling a free group along various infinitely generated subgroups, expanding understanding of such groups' diversity.

Contribution

It introduces a method to generate uncountably many non-isomorphic finitely generated residually finite groups through free group doubling along infinitely generated subgroups.

Findings

Uncountably many non-isomorphic groups constructed

Groups are finitely generated and residually finite

Method demonstrates vast diversity in such group families

Abstract

We study a family of finitely generated residually finite groups. These groups are doubles $F_2*_H F_2$ of a rank-$2$ free group $F_2$ along an infinitely generated subgroup $H$. Varying $H$ yields uncountably many groups up to isomorphism.