Polynomial time conjugacy in wreath products and free solvable groups

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.