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CSA-groups and separated free constructions. (English) Zbl 0838.20025
A group $$G$$ is called a CSA-group if all maximal Abelian subgroups are malnormal; i.e. $$M^x\cap M=1$$ for every maximal Abelian subgroup $$M$$ and $$x\in G\setminus M$$. The class of CSA-groups contains all torsion free hyperbolic groups and all groups freely acting on $$\Lambda$$-trees. The authors describe conditions under which HNN-extensions and amalgamated free products of CSA-groups are again CSA. One relator CSA-groups are characterised as follows in the paper: a torsion free one-relator group is CSA if and only if it does not contain $$F_2\times\mathbb{Z}$$ or one of the metabelian Baumslag-Solitar groups $$B_{1,n}=\langle x,y\mid yxy^{-1}=x^n\rangle$$, $$n\in \mathbb{Z}\setminus \{0,1\}$$; a one-relator group with torsion is CSA if and only if it does not contain the infinite dihedral group.
Reviewer: P.Zalesskij (Wien)

##### MSC:
 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20F05 Generators, relations, and presentations of groups 20E07 Subgroup theorems; subgroup growth
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