Finitely generated groups with the M. Hall property
type: Item
instance of: scholarly article
title: Finitely generated groups with the M. Hall property
zbMATH Open document ID: 0836.20026
DOI: 10.1007/BF02259853
author: Oleg Bogopolski
published in: Algebra and Logic
publication date: 1994-04-12T00:00:00Z
review text: An M. Hall group is one in which every finitely generated subgroup is a free factor of a subgroup of finite index. In particular, free groups are M. Hall. Stallings' theory of ends of groups yields readily that a finitely generated, accessible M. Hall group is isomorphic to the fundamental group of a finite graph of finite groups. Write \(\mathcal J\) for the class of groups isomorphic to such fundamental groups. It is natural to ask which groups in \(\mathcal J\) are Hall. It was claimed in the article [Can. J. Math. 31, 1329-1338 (1979; Zbl 0393.20017)] by \textit{A. M. Brunner} and the reviewer, that an amalgamated product \(A *_U B\) with \(A\), \(B\) finite, is M. Hall if and only if \(U\) is malnormal in at least one of \(A\), \(B\). The present author shows by means of a counterexample that this condition is in fact not necessary (though it is sufficient) and locates the error in the putative proof. It is easy to see that any M. Hall group \(G\) has the property \(\mathcal N\): for every non-trivial finitely generated subgroup \(H < G\), the index \(|N_G (H) : H|< \infty\). Using covering-space arguments involving a 3-complex with fundamental group isomorphic to \(G \in {\mathcal J}\), a criterion is found for \(G\) to be M. Hall, which moreover can be checked algorithmically. It is also shown that the question as to whether a group in \(\mathcal J\) has property \(\mathcal N\) is algorithmically decidable. Necessary and sufficient conditions are given for a group in \(\mathcal J\) to have a free subgroup of given rank and (finite) index. Finally, a counterexample is given to the conjecture of Brunner and the reviewer (see the reference above) that a subgroup in \(\mathcal J\) is M. Hall if and only if it has property \(\mathcal N\).
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F34
zbMATH DE Number: 536662
zbMATH Keywords: covering spaces
zbMATH Keywords: finitely generated subgroups
zbMATH Keywords: free factors
zbMATH Keywords: subgroup of finite index
zbMATH Keywords: free groups
zbMATH Keywords: ends of groups
zbMATH Keywords: accessible M. Hall groups
zbMATH Keywords: finite graph of finite groups
zbMATH Keywords: fundamental groups
zbMATH Keywords: amalgamated products
zbMATH Keywords: M. Hall groups
MaRDI profile type: Publication
cites work: Q5551146
cites work: Autostability of models
cites work: Recursive Labelling Systems and Stability of Recursive Structures in Hyperarithmetical Degrees
cites work: Q5573961
Recommended article: Almost free groups and the M. Hall property
Recommended article: On Certain Finitely Generated Subgroups of Groups Which Split
Recommended article: Q4942281
Recommended article: MALNORMALITY IS DECIDABLE IN FREE GROUPS
Recommended article: On Some Finiteness Properties in Infinite Groups
Recommended article: Virtually free factors of pro-\(p\) groups
Recommended article: On residually finite graph products
Recommended article: Q4344649
Recommended article: The free product of groups with amalgamated subgroup malnormal in a single factor
Recommended article: Subgroup Separability of Generalized Free Products of Free-By-Finite Groups
description: scientific article; zbMATH DE number 536662