CSA-groups and separated free constructions

A group G is called a CSA-group if all its maximal Abelian subgroups are malnormal; that is, M^x ∩ M = 1 for every maximal Abelian subgroup M and x ∩ G − M. The class of CSA-groups contains all torsion-free hyperbolic groups and all groups freely acting on λ-trees. We describe conditions under which HNN-extensions and amalgamated products of CSA-groups are again CSA. One-relator CSA-groups are characterised as follows: a torsion-free one-relator group is CSA if and only if it does not contain F₂ × Z or one of the nonabelian metabelian Baumslag-Solitar groups B_{1, n} = 〈x, y | yxy⁻¹ = xⁿ〉, n ∈ Z ∂ {0, 1}; a one-relator group with torsion is CSA if and only if it does not contain the infinite dihedral group.