Subgroups of groups finitely presented in Burnside varieties

Abstract

For all sufficiently large odd integers $n$, the following version of Higman's embedding theorem is proved in the variety ${\mathcal{B}}_n$ of all groups satisfying the identity $x^n=1$. A finitely generated group $G$ from ${\mathcal{B}}_n$ has a presentation $G=⟨A\mid R⟩$ with a finite set of generators $A$ and a recursively enumerable set $R$ of defining relations if and only if it is a subgroup of a group $H$ finitely presented in the variety ${\mathcal{B}}_n$. It follows that there is a 'universal' $2$-generated finitely presented in ${\mathcal{B}}_n$ group containing isomorphic copies of all finitely presented in ${\mathcal{B}}_n$ groups as subgroups.

84 pages, 14 figures