The Conjugacy Problem in Free Solvable Groups and Wreath Products of Abelian Groups is in TC0

  • Alexei Miasnikov
  • Svetla Vassileva
  • Armin WeißEmail author
Part of the following topical collections:
  1. Computer Science Symposium in Russia


We show that the conjugacy problem in a wreath product AB is uniform-TC0-Turing-reducible to the conjugacy problem in the factors A and B and the power problem in B. If B is torsion free, the power problem in B can be replaced by the slightly weaker cyclic submonoid membership problem in B. Moreover, if A is abelian, the cyclic subgroup membership problem suffices, which itself is uniform-AC0-many-one-reducible to the conjugacy problem in AB. Furthermore, under certain natural conditions, we give a uniform TC0 Turing reduction from the power problem in AB to the power problems of A and B. Together with our first result, this yields a uniform TC0 solution to the conjugacy problem in iterated wreath products of abelian groups – and, by the Magnus embedding, also in free solvable groups.


Wreath products Conjugacy problem Word problem TC0 Free solvable group 


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Authors and Affiliations

  1. 1.Stevens Institute of TechnologyHobokenUSA
  2. 2.Champlain CollegeSt-LambertCanada
  3. 3.Universität StuttgartStuttgartGermany

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