, Volume 48, Issue 2, pp 89-98
Date: 06 Jun 2009

The twisted conjugacy problem for endomorphisms of metabelian groups

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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 φ ∈ End(M) identical modulo an Abelian normal subgroup N containing the derived subgroup M′ and for any pair of elements u, vM, decides if an equation of the form ()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 φ ∈ End(M).