Conjugacy Distinguished Subgroups

TL;DR

This paper introduces the concept of conjugacy ${ m C}$-distinguished subgroups in residually ${ m C}$ groups and proves their prevalence in certain classes of groups, including limit and Lyndon groups.

Contribution

It establishes that finitely generated subgroups are conjugacy ${ m C}$-distinguished in groups with specific normal free subgroups and extends this property to limit, Lyndon, and certain one-relator groups.

Findings

Finitely generated subgroups are conjugacy ${ m C}$-distinguished in groups with a normal free subgroup and quotient in ${ m C}$.

Finitely generated subgroups of limit groups are conjugacy ${ m C}$-distinguished.

Finitely generated subgroups of Lyndon and certain one-relator groups are conjugacy ${ m C}$-distinguished.

Abstract

Let ${\cal C}$ be a nonempty class of finite groups closed under taking subgroups, homomorphic images and extensions. A subgroup $H$ of an abstract residually ${\cal C}$ group $R$ is said to be conjugacy ${\cal C}$-distinguished if whenever $y\in R$, then $y$ has a conjugate in $H$ if and only if the same holds for the images of $y$ and $H$ in every quotient group $R/N\in {\cal C}$ of $R$. We prove that in a group having a normal free subgroup $\Phi$ such that $R/\Phi$ is in ${\cal C}$, every finitely generated subgroup is conjugacy ${\cal C}$-distinguished. We also prove that finitely generated subgroups of limit groups, of Lyndon groups and certain one-relator groups are conjugacy distinguished (${\cal C}$ here is the class of all finite groups).