Limit Groups are Subgroup Conjugacy Separable

TL;DR

This paper proves that limit groups, certain one relator groups, and virtual retracts of hyperbolic virtually special groups are subgroup conjugacy separable, meaning their non-conjugate finitely generated subgroups can be distinguished in finite quotients.

Contribution

It establishes subgroup conjugacy separability for limit groups, specific one relator groups, and virtual retracts of hyperbolic virtually special groups, expanding understanding of subgroup separability properties.

Findings

Limit groups are subgroup conjugacy separable.

Certain one relator groups are subgroup conjugacy separable.

Virtual retracts of hyperbolic virtually special groups are subgroup conjugacy separable.

Abstract

A group $G$ is called subgroup conjugacy separable if for every pair of non-conjugate finitely generated subgroups of $G$, there exists a finite quotient of $G$ where the images of these subgroups are not conjugate. We prove that limit groups are subgroup conjugacy separable. We also prove this property for one relator groups of the form $R=\langle a_1,...,a_n\mid W^n\rangle$ with $n>|W|$. The property is also proved for virtual retracts (equivalently for quasiconvex subgroups) of hyperbolic virtually special groups.