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Contents
1. Introduction
2. Preliminaries
3. Step 1: A rigidity theorem
4. Step 2: Generalised loxodromic elements
5. Step 3: Fixators in the outer automorphism group
6. Step 4: Fixed point property on real trees
7. Proofs of the main theorem and its corollaries
8. A few other applications and some open questions
A. A general hyperbolic model: the graph of maximal products



