E. Ventura and V. A Roman’kov (2009)

# The twisted conjugacy problem for endomorphisms of metabelian groups

Algebra and Logic, 48(2):89–98.

Let M be a finitely generated metabelian group explicitly presented in a variety A2 \mathcal\A\\ˆ2 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, v ∈ M, decides if an equation of the form (xφ)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).

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