A Bers-like proof of the existence of train tracks for free group automorphisms

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Using Lipschitz distance on Outer space we give another proof of the train track theorem.

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Constructing group actions on quasi-trees and applications to mapping class groups (2010, background): “rmation on Out(Fn) and Outer space. An element f of Out(Fn) is fully irreducible if there are no conjugacy classes of proper free factors of Fnwhich are f-periodic. Such elements have axes in CV, see [Bes11]. In [AK11] Yael Algom-Kﬁr shows that there is ν>0 such that the projection of any transla”
Hyperbolicity of the complex of free factors (2011, background): “′ is the log of the minimal Lipschitz constant over all diﬀerence of markings maps G→ G′. The Lipschitz distance is an asymmetric metric on CV inducing the correct topology. For more information, see [11, 2, 4]. The basic fact, that plays the role of Teichmu¨ller’s theorem in Teichmu¨ller theory, is”
Asymmetry of Outer space (2009, methodology): “ace by isometries, and if Φ ∈ Out(Fn) is a fully irreducible automorphism then the translation distance of Φ equals logλ, the growth rate of Φ. This last property was exploited to give a new proof in [Bes] of the train track theorem [BH92]. Moreover, Φ acts as a translation by logλ on certain biinﬁn”