Isomorphism types in wreath products and effective embeddings of periodic groups

Abstract:

For any finitely generated group $Y,\omega (Y)$ 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 $k \geqslant 1$ elements. We prove that if $\tau$ is any Turing degree with $\tau \geqslant 1.{\text {u.b.}}\{ {\omega (G),\omega (B)} \}$ then the unrestricted wreath product $W = G{\text {Wr}} B$ has a $( {k + 1} )$-generator subgroup $H$ with $\omega (H) = \tau$. If $B$ is also periodic, then $W$ has a $k$-generator subgroup $H$ such that $\tau = 1.{\text {u.b.}}\{ {\omega (B),\omega (H)} \}$. Easy consequences include: $G{\text {Wr}} {\mathbf {Z}}$ has ${2^{{\aleph _0}}}$ pairwise nonembeddable $2$-generator subgroups and if $B$ is periodic then $G{\text {Wr}} B$ has ${2^{{\aleph _0}}}$ pairwise nonembeddable $k$-generator subgroups. Using similar methods, we prove an effective embedding theorem for embedding countable periodic groups in $2$-generator periodic groups.