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

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Abstract

We show that the conjugacy problem in a wreath product A ≀ B 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 A ≀ B. Furthermore, under certain natural conditions, we give a uniform TC0 Turing reduction from the power problem in A ≀ B 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.

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Authors and Affiliations

  1. Stevens Institute of Technology, Hoboken, NJ, USA

    Alexei Miasnikov

  2. Champlain College, St-Lambert, QC, Canada

    Svetla Vassileva

  3. Universität Stuttgart, Stuttgart, Germany

    Armin Weiß

Authors
  1. Alexei Miasnikov
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  2. Svetla Vassileva
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  3. Armin Weiß
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Correspondence to Armin Weiß.

Additional information

This article is part of the Topical Collection on Computer Science Symposium in Russia

The results of this paper were obtained with the support of the Russian Science Foundation (project No. 17-11-0117). Most parts of this research have been conducted while the third author was at Stevens Institute of Technology. In Stuttgart he was partially supported by the DFG-project: DI 435/7-1 “Algorithmic problems in group theory”.

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Miasnikov, A., Vassileva, S. & Weiß, A. The Conjugacy Problem in Free Solvable Groups and Wreath Products of Abelian Groups is in TC0. Theory Comput Syst 63, 809–832 (2019). https://doi.org/10.1007/s00224-018-9849-2

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