The twisted conjugacy problem for endomorphisms of metabelian groups Authors E. Ventura University Politécnica de Catalunya V. A. Roman’kov Dostoevskii Omsk State University Article

First Online: 06 June 2009 Received: 25 December 2008 DOI :
10.1007/s10469-009-9048-y

Cite this article as: Ventura, E. & Roman’kov, V.A. Algebra Logic (2009) 48: 89. doi:10.1007/s10469-009-9048-y 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 , 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 ).

Keywords metabelian group twisted conjugacy endomorphism fixed points Fox derivatives Supported by RFBR (project No. 07-01-00392). (V. A. Roman’kov)

Translated from Algebra i Logika , Vol. 48, No. 2, pp. 157–173, March–April, 2009.

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