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For any finitely generated group Y, w(Y) denotes the Turing degree of the word problem of Y. Let G be any non-Abelian 2-generator group and B an infinite group generated by k : I elements. We prove that if T is any Turing degree with T : l.u.b.{w(G), w(B)} then the unrestricted wreath product W = GWr B has a (k + I)-generator subgroup H with w(H) = T. If B is also periodic, then W has a k-generator subgroup H such that T = l.u.b.{w(B), w(H)}. Easy consequences include: G Wr Z has 28o pairwise nonembeddable 2-generator subgroups and if B is periodic then GWr B has 28o pairwise nonembeddable k-generator subgroups. Using similar methods, we prove an effective embedding theorem for embedding countable periodic groups in 2-generator periodic groups.

Isomorphism types in wreath products and effective embeddings of periodic groups