TAILIEUCHUNG - Đề tài " The Tits alternative for Out(Fn) II: A Kolchin type theorem "

This is the second of two papers in which we prove the Tits alternative for Out(Fn ). Contents 1. Introduction and outline 2. Fn -trees . Real trees . Real Fn -trees . Very small trees . Spaces of real Fn -trees . Bounded cancellation constants . Real graphs . Models and normal forms for simplicial Fn -trees . Free factor systems 3. Unipotent polynomially growing outer automorphisms . Unipotent linear maps . Topological representatives . Relative train tracks and automorphisms of polynomial growth . Unipotent representatives and UPG automorphisms . | Annals of Mathematics The Tits alternative for Out Fn II A Kolchin type theorem By Mladen Bestvina Mark Feighn and Michael Handel Annals of Mathematics 161 2005 1 59 The Tits alternative for Out Fn II A Kolchin type theorem By Mladen Bestvina Mark Feighn and Michael Handel Abstract This is the second of two papers in which we prove the Tits alternative for Out Fn . Contents 1. Introduction and outline 2. - -trees . Real trees . Real Fn-trees . Very small trees . Spaces of real Fn-trees . Bounded cancellation constants . Real graphs . Models and normal forms for simplicial Fn-trees . Free factor systems 3. Unipotent polynomially growing outer automorphisms . Unipotent linear maps . Topological representatives . Relative train tracks and automorphisms of polynomial growth . Unipotent representatives and UPG automorphisms 4. The dynamics of unipotent automorphisms . Polynomial sequences . Explicit limits . Primitive subgroups . Unipotent automorphisms and trees 5. A Kolchin theorem for unipotent automorphisms . F contains the suffixes of all nonlinear edges . Bouncing sequences stop growing . Bouncing sequences never grow The authors gratefully acknowledge the support of the National Science Foundation. 2 MLADEN BESTVINA MARK FEIGHN AND MICHAEL HANDEL . Finding Nielsen pairs . Distances between the vertices . Proof of Theorem 6. Proof of the main theorem References 1. Introduction and outline Recent years have seen a development of the theory for Out Fn the outer automorphism group of the free group Fn of rank n that is modeled on Nielsen-Thurston theory for surface homeomorphisms. As mapping classes have either exponential or linear growth rates so free group outer automorphisms have either exponential or polynomial growth rates. The degree of the polynomial can be any integer between 1 and n 1 see BH92 . In BFH00 we considered individual automorphisms with primary emphasis on those with .

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