Isomorphism Types in Wreath Products and Effective Embeddings of Periodic Groups

For any finitely generated group Y, u(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 > 1 elements. We prove that if t is any Turing degree with t > l.u.b.(co(G), u(ß)} then the unrestricted wreath product W = GWrB has a (k -f l)-generator subgroup H with u(H) = r. If B is also periodic, then W has a /c-generator subgroup //such thatT = l.u.b.{w(B), «(//)}. Easy consequences include: GWrZ has 2S° pairwise nonembeddable 2-generator subgroups and if B is periodic then GWr/3 has 2K° pairwise nonembeddable ^-generator subgroups. Using similar methods, we prove an effective embedding theorem for embedding countable periodic groups in 2-generator periodic groups.