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Howson property and one-relator groups. (English) Zbl 0922.20046
A group \(G\) is said to have the Howson property if the intersection of any two finitely generated subgroups of \(G\) is again finitely generated. There is a large literature on Howson groups, and the question of which one-relator subgroups have the Howson property is an interesting one. All the examples previous to this paper, of non-Howson finitely generated one-relator groups, are not word hyperbolic. In this paper, the author gives an example of a torsion-free one-relator group \(G\) which is word hyperbolic and which does not have the Howson property. Moreover, \(G\) is not subgroup separable and it contains a free subgroup of rank two that is not quasi-convex in \(G\). The author discusses then similarities and differences with 3-manifold groups and he presents two conjectures related to his results.

MSC:
20F65 Geometric group theory
20E07 Subgroup theorems; subgroup growth
20F05 Generators, relations, and presentations of groups
Software:
MAGNUS
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