On Benign Subgroups Constructed by Higman’s Sequence Building Operation

Pub Date : 2024-04-09 DOI:10.3103/s1068362324010023

V. S. Atabekyan, V. H. Mikaelian

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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.

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论通过希格曼序列构建操作构建的良性子群

摘要对于希格曼的序列建立操作 \(\omega_{m}\)和任何整数序列集 \({\mathcal{B}}}\)，只要 \(A_{\omega_{m}{mathcal{B}}}\)在自由群 \(G\)中是良性的，子群 \(A_{\mathcal{B}}\)在 \(G\)中就是良性的。希格曼利用这一性质作为证明有限生成群可嵌入有限呈现群的关键步骤，当且仅当它是递归呈现的。我们建立了这一事实的显式类比，即、我们明确地给出了一个有限呈现过群 \(K_{\omega_{m} {mathcal{B}}}\) of \(G\)和它的有限生成子群\L_{omega_{m}{mathcal{B}}}leq K_{omega_{m}{mathcal{B}}}\) 这样 \(G\cap L_{omega_{m}{mathcal{B}}=A_{omega_{m}{mathcal{B}}}\) 成立。我们的构造可以用于有限生成群到有限呈现群的显式嵌入，希格曼定理在理论上是可行的。为了建立我们的构造，我们提出了一些辅助的''嵌套''自由构造，它们基于自由积与合并和 HNN 扩展。

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