On generalized (twisted) conjugacy separability of some extensions of groups

TL;DR

This paper explores generalized conjugacy separability properties in group extensions, establishing equivalences and implications among various conjugacy problems, with applications to specific classes of groups.

Contribution

It introduces new separability properties related to conjugacy and twisted conjugacy, and proves their equivalences in finite and cyclic group extensions.

Findings

Generalized twisted conjugacy separability is equivalent to conjugacy problem in finite extensions.

Conjugacy separability in cyclic extensions implies twisted conjugacy separability.

Virtually free times free groups are proven to be conjugacy separable.

Abstract

We introduce separability properties corresponding to generalized versions of the conjugacy, twisted conjugacy, Brinkmann and Brinkmann's conjugacy problems and how they relate when finite and cyclic extensions of groups are taken. In particular, we prove that some (concrete) generalizations of twisted conjugacy separability of a group $G$ with respect to virtually inner automorphisms are equivalent to some (concrete) generalizations of the conjugacy problem in finite extensions of $G$. Similarly, (generalized) conjugacy separability in cyclic extensions of $G$ implies (generalized) twisted conjugacy and Brinkmann's conjugacy separability in $G$. Applications include results in free, virtually abelian, virtually polycyclic groups and a proof that virtually free times free groups are conjugacy separable.