The twisted conjugacy problem for pairs of endomorphisms in nilpotent groups

TL;DR

This paper presents algorithms to decide the twisted conjugacy problem for pairs of endomorphisms in finitely generated nilpotent groups and to compute the equalizer subgroup, establishing the problem's decidability.

Contribution

It introduces the first decidability algorithms for the twisted conjugacy problem in finitely generated nilpotent groups and provides methods to compute related subgroups.

Findings

The twisted conjugacy problem is decidable for finitely generated nilpotent groups.

Algorithms are provided to find solutions to the twisted conjugacy equation.

A finite generating set for the equalizer subgroup is computable.

Abstract

An algorithm is constructed that, when given an explicit presentation of a finitely generated nilpotent group $G,$ decides for any pair of endomorphisms $\varphi, \psi : G \to G$ and any pair of elements $u, v \in G,$ whether or not the equation $(x\varphi)u = v (x\psi)$ has a solution $x \in G.$ Thus it is shown that the problem of the title is decidable. Also we present an algorithm that produces a finite set of generators of the subgroup (equalizer) $Eq_{\varphi, \psi}(G) \leq G$ of all elements $u \in G$ such that $u \varphi = u \psi .$