  We prove that the complexity of the Conjugacy Problems for wreath products
and for free solvable groups is decidable in polynomial time. For the wreath
product AwrB, we must assume the decidability in polynomial time of the
Conjugacy Problems for A and B and of the power problem in B. We obtain the
result by making the algorithm for the Conjugacy Problem described in a paper
of Matthews run in polynomial time. Using this result and properties of the
Magnus embedding, we show that the Conjugacy and Conjugacy Search Problems in
free solvable groups are computable in polynomial time.
