The Conjugacy Problem in Free Solvable Groups and Wreath Products of Abelian Groups is in \({{\mathsf {T}}}{{\mathsf {C}}}^0\)

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10304)

Abstract

We show that the conjugacy problem in a wreath product \(A \wr B\) is uniform-\({{\mathsf {T}}}{{\mathsf {C}}}^0\)-Turing-reducible to the conjugacy problem in the factors A and B and the power problem in B. Moreover, if B is torsion free, the power problem for B can be replaced by the slightly weaker cyclic submonoid membership problem for B, which itself turns out to be uniform-\({{\mathsf {T}}}{{\mathsf {C}}}^0\)-Turing-reducible to the conjugacy problem in \(A \wr B\) if A is non-abelian.

Furthermore, under certain natural conditions, we give a uniform \({{\mathsf {T}}}{{\mathsf {C}}}^0\) Turing reduction from the power problem in \(A \wr B\) to the power problems of A and B. Together with our first result, this yields a uniform \({{\mathsf {T}}}{{\mathsf {C}}}^0\) solution to the conjugacy problem in iterated wreath products of abelian groups – and, by the Magnus embedding, also for free solvable groups.

Keywords

Wreath products Conjugacy problem Word problem \({{\mathsf {T}}}{{\mathsf {C}}}^0\) Free solvable group 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Alexei Miasnikov
    • 1
  • Svetla Vassileva
    • 2
  • Armin Weiß
    • 1
  1. 1.Stevens Institute of TechnologyHobokenUSA
  2. 2.Champlain CollegeSt-lambertCanada

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