The conjugacy problem in free solvable groups and wreath product of abelian groups is in TC$^0$

We show that the conjugacy problem in a wreath product A ≀B is uniformTC -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-TC-Turing-reducible to conjugacy problem in A ≀B if A is non-abelian. Furthermore, under certain natural conditions, we give a uniform TC 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 TC solution to the conjugacy problem in iterated wreath products of abelian groups – and, by the Magnus embedding, also for free solvable groups.