We show that every two-generated torsion-free one-ended word-hyperbolic group has virtually cyclic outer automorphism group. This is done by computing the JSJ-decomposition for two-generator hyperbolic groups. We further prove that two-generated torsion-free word-hyperbolic groups are strongly accessible. This means that they can be constructed from groups with no nontrivial cyclic splittings by applying finitely many free products with amalgamation and HNN-extensions over cyclic subgroups.
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