On Benign Subgroups Constructed by Higman’s Sequence Building Operation

Abstract

For Higman’s sequence building operation \(\omega_{m}\) and for any integer sequences set \({\mathcal{B}}\) the subgroup \(A_{\omega_{m}{\mathcal{B}}}\) is benign in a free group \(G\) as soon as \(A_{\mathcal{B}}\) is benign in \(G\). Higman used this property as a key step to prove that a finitely generated group is embeddable into a finitely presented group if and only if it is recursively presented. We build the explicit analog of this fact, i.e., we explicitly give a finitely presented overgroup \(K_{\omega_{m}{\mathcal{B}}}\) of \(G\) and its finitely generated subgroup \(L_{\omega_{m}{\mathcal{B}}}\leq K_{\omega_{m}{\mathcal{B}}}\) such that \(G\cap L_{\omega_{m}{\mathcal{B}}}=A_{\omega_{m}{\mathcal{B}}}\) holds. Our construction can be used in explicit embeddings of finitely generated groups into finitely presented groups, which are theoretically possible by Higman’s theorem. To build our construction we suggest some auxiliary ‘‘nested’’ free constructions based on free products with amalgamation and HNN-extensions.