Subgroup Conjugacy Separability in Residually Free Groups

TL;DR

This work addresses subgroup conjugacy separability in residually free groups and shows that finitely presented residually free groups possess this property; moreover, those of type $FP_\infty$ are subgroup conjugacy distinguished. By leveraging a connection between conjugacy separability and residual finiteness of outer automorphism groups (via Grossman), the authors prove that $Out(G)$ is residually finite for finitely presented residually free groups. The methodology combines embeddings into finite direct products of limit groups with profinite-topology arguments and virtual-retract techniques to lift conjugacy properties from ambient products to the group. The results extend prior understanding of limit groups, imply decidability consequences for subgroups, and rule out certain nontrivial Baumslag–Solitar subgroups in this context.

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.

Theorems & Definitions (20)

• Theorem
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Definition 3.1
• Proposition 3.2
• proof