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.
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Miasnikov, A., Vassileva, S., Weiß, A. (2017). The Conjugacy Problem in Free Solvable Groups and Wreath Products of Abelian Groups is in \({{\mathsf {T}}}{{\mathsf {C}}}^0\) . In: Weil, P. (eds) Computer Science – Theory and Applications. CSR 2017. Lecture Notes in Computer Science(), vol 10304. Springer, Cham. https://doi.org/10.1007/978-3-319-58747-9_20
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