Conjugacy Distinguished Subgroups

Abstract

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

Section 3 is improved, the main results are unchanged