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We examine the action of Out(F n ) on the set of (irreducible) laminations. Consequences include a special case of the Tits alternative for Out(F n ), the discreteness of certain naturally arising group actions on trees, and word hyperbolicity of certain semidirect products. Introduction and PreliminariesThe outer automorphism group Out(F n ) of the free group F n of rank n naturally maps onto GL n (Z) and contains as a subgroup the mapping class group of a compact surface with fundamental group F n . It should not be surprising then to expect Out(F n ) to exhibit the phenomena present in both linear groups and mapping class groups. Much of the recent research of Out(F n ) has focused on developing tools and proving results known in the other two categories. Culler and Vogtmann have constructed in [CuV] a contractible complex "Outer Space" on which Out(F n ) acts discretely and with finite stabilizers. This space plays the role of the symmetric space and of the Teichmüller space. As a consequence, Culler and Vogtmann conclude that Out(F n ) has type V F L, i.e. a subgroup G of finite index has a K(G, 1) with finitely many cells, thus generalizing the corresponding results for GL n (Z) and mapping class groups. In [CuMo] Culler and Morgan construct an equivariant compactification of Outer Space, much like Thurston's compactification of Teichmüller space. A point in compactified Outer Space represents an action of F n on a real tree and these in turn play the role of measured geodesic laminations on surfaces.In [BeH] "train-track representatives" of outer automorphisms were constructed. These are analogs of the Jordan normal form and the NielsenThurston normal form. They were used to prove that the fixed subgroup of an automorphism of F n has rank ≤ n.

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