Geodesic
The twisted conjugacy problem for endomorphisms of metabelian groups
Algebra i logika, Tome 48 (2009) no. 2, pp. 157-173

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Let $M$ be a finitely generated metabelian group explicitly presented in a variety $\mathcal A^2$ of all metabelian groups. An algorithm is constructed which, for every endomorphism $\varphi\in\operatorname{End}(M)$ identical modulo an Abelian normal subgroup $N$ containing the derived subgroup $M'$ and for any pair of elements $u,v\in M$, decides if an equation of the form $(x\varphi)u=vx$ has a solution in $M$. Thus, it is shown that the title problem under the assumptions made is algorithmically decidable. Moreover, the twisted conjugacy problem in any polycyclic metabelian group $M$ is decidable for an arbitrary endomorphism $\varphi\in\operatorname{End}(M)$.
Keywords: metabelian group, twisted conjugacy, fixed points, Fox derivatives.
Mots-clés : endomorphism 
@article{AL_2009_48_2_a0,
     author = {E. Ventura and V. A. Roman'kov},
     title = {The twisted conjugacy problem for endomorphisms of metabelian groups},
     journal = {Algebra i logika},
     pages = {157--173},
     publisher = {mathdoc},
     volume = {48},
     number = {2},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2009_48_2_a0/}
}
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E. Ventura; V. A. Roman'kov. The twisted conjugacy problem for endomorphisms of metabelian groups. Algebra i logika, Tome 48 (2009) no. 2, pp. 157-173. http://geodesic.mathdoc.fr/item/AL_2009_48_2_a0/