Subgroup Conjugacy Separability in Residually Free Groups

TL;DR

This paper proves that finitely presented residually free groups are subgroup conjugacy separable and, under certain conditions, also subgroup conjugacy distinguished, with implications for their outer automorphism groups.

Contribution

It establishes subgroup conjugacy separability for finitely presented residually free groups and links conjugacy separability to residual finiteness of their outer automorphism groups.

Findings

Finitely presented residually free groups are subgroup conjugacy separable.

Such groups are subgroup conjugacy distinguished if of type FP_infinity.

Their outer automorphism groups are residually finite.

Abstract

We prove that finitely presented residually free groups are subgroup conjugacy separable. Furthermore, if they are of type $FP_\infty$, then they are also subgroup conjugacy distinguished. Using a connection between conjugacy separability and residual finiteness of outer automorphism group established by Grossman in \cite{Grossman}, we show that finitely presented residually free groups have residually finite outer automorphism groups.