Finitely generated groups with the M. Hall property
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 \(\mathcal N\).
scientific article; zbMATH DE number 536662