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An inaccessible group. (English) Zbl 0833.20035
Niblo, Graham A. (ed.) et al., Geometric group theory. Volume 1. Proceedings of the symposium held at the Sussex University, Brighton (UK), July 14-19, 1991. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 181, 75-78 (1993).
A finitely generated group is said to be accessible if it is the fundamental group of a graph of groups in which all edge groups are finite and every vertex group has at most one end. M. Bestvina and M. Feighn [in: Arboreal group theory, Publ., Math. Sci. Res. Inst. 19, 133-141 (1991; Zbl 0826.20027)] gave an example of a finitely generated inaccessible group which does not satisfy a generalized accessibility condition in which decomposition over torus subgroups is allowed. In the paper under review, the author gives an elegant example of an inaccessible group. The idea is to use a certain lattice of subgroups containing an increasing sequence of finite groups \(H_1\subset H_2\subset\cdots\), with each \(H_i\) contained in a larger finite group \(K_i\), and infinite groups \(G_{i+1}\) generated by \(K_i\) and \(H_{i+1}\). Let \(P\) be the infinite amalgamated free product \(P=G_1*_{H_1}G_2*_{H_2}G_3\cdots\), let \(H\) be a finitely generated group which contains \(H_\omega=\bigcup^\infty_{i =1}H_i\), and put \(J=P*_{H_\omega} H\). An inductive argument shows that \(J\) is generated by \(G_1\) and \(H\), hence is finitely generated. Since \(J\) can be decomposed as a free product with amalgamation \(G_1*_{K_1}G_2*\cdots G_n*_{K_n}J_n\) for arbitrarily large \(n\), it is not accessible. The proof is completed by giving an explicit construction of a lattice of groups with the required properties.
For the entire collection see [Zbl 0777.00044].

20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F05 Generators, relations, and presentations of groups
20E15 Chains and lattices of subgroups, subnormal subgroups