On the structure of two-generated hyperbolic groups

We show that every two-generated torsion-free one-ended word-hyperbolic gro up has virtually cyclic outer automorphism group. This is done by computing the JSJ-decomposition for two-generator hyperbolic groups. We further prov e that two-generated torsion-free word-hyperbolic groups are strongly acces sible. This means that they can be constructed from groups with no nontrivi al cyclic splittings by applying finitely many free products with amalgamat ion and HNN-extensions over cyclic subgroups.