On generalized conjugacy and some related problems

TL;DR

This paper explores the connections between generalized conjugacy problems in certain groups and algorithmic problems, establishing decidability results for specific classes like virtually polycyclic and free groups.

Contribution

It links the generalized conjugacy problem for G-by-Z groups to algorithmic problems in G, proving decidability for virtually polycyclic and free groups.

Findings

Decidability of GBrCP(G) for virtually polycyclic groups.

Decidability of GBrP(G) for finitely generated abelian groups.

Decidability of BrP(G) and TCP(G) for finitely generated virtually free groups.

Abstract

We establish a connection between the generalized conjugacy problem for a $G$-by-$\mathbb{Z}$ group, $GCP(G \rtimes \mathbb{Z})$, and two algorithmic problems for $G$: the generalized Brinkmann's conjugacy problem, $GBrCP(G)$, and the generalized twisted conjugacy problem, $GTCP(G)$. We explore this connection for generalizations of different kinds: relative to finitely generated subgroups, to theirs cosets, or to recognizable, rational, context-free or algebraic subsets of the group. Using this result, we are able to prove that $GBrCP(G)$ is decidable (with respect to cosets) when $G$ is a virtually polycyclic group, which implies in particular that the generalized Brinkmann's equality problem, $GBrP(G)$, is decidable if $G$ is a finitely generated abelian group. Finally, we prove that if $G$ is a finitely generated virtually free group, then the simple versions of Brinkmann's equality problem and of the twisted conjugacy problem, $BrP(G)$ and $TCP(G)$, are decidable.