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On the Separability of Abelian Subgroups of the Fundamental Groups of Graphs of Groups. II

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Abstract

Consider the fundamental group \( {\mathfrak{G}} \) of an arbitrary graph of groups and some root class \( {\mathcal{C}} \) of groups, i.e., a class containing a nontrivial group and closed under subgroups, extensions, and unrestricted direct products of the form \( \prod_{y\in Y}X_{y} \), where \( X,Y\in{\mathcal{C}} \) and \( X_{y} \) is an isomorphic copy of \( X \) for each \( y\in Y \). We provide some criterion for the separability by \( {\mathcal{C}} \) of a finitely generated abelian subgroup of \( {\mathfrak{G}} \) valid when the group satisfies an analog of the Baumslag filtration condition. This enables us to describe the \( {\mathcal{C}} \)-separable finitely generated abelian subgroups for the fundamental groups of some graphs of groups with central edge subgroups.

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Funding

The study was supported by the Russian Science Foundation grant no. 22–21–00166, https://rscf.ru/en/project/22-21-00166/.

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  1. Ivanovo State University, Ivanovo, Russia

    E. V. Sokolov

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  1. E. V. Sokolov
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Correspondence to E. V. Sokolov.

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Translated from Sibirskii Matematicheskii Zhurnal, 2024, Vol. 65, No. 1, pp. 207–228. https://doi.org/10.33048/smzh.2024.65.116

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Sokolov, E.V. On the Separability of Abelian Subgroups of the Fundamental Groups of Graphs of Groups. II. Sib Math J 65, 174–189 (2024). https://doi.org/10.1134/S0037446624010166

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  • DOI: https://doi.org/10.1134/S0037446624010166

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