The twisted conjugacy problem for pairs of endomorphisms in nilpotent groups

Abstract

An algorithm is constructed that, when given an explicit presentation of a finitely generated nilpotent group $G,$ decides for any pair of endomorphisms $\phi ,\psi :G\to G$ and any pair of elements $u,v\in G,$ whether or not the equation $(x\phi )u=v(x\psi )$ 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) $E{q}_{\phi ,\psi }(G)\le G$ of all elements $u\in G$ such that $u\phi =u\psi .$