Stretching factors, metrics and train tracks for free products

Abstract

In this paper, we develop the metric theory for the outer space of a free product of groups. This generalizes the theory of the outer space of a free group, and includes its relative versions. The outer space of a free product is made of $G$-trees with possibly non-trivial vertex stabilisers. The strategies are the same as in the classical case, with some technicalities arising from the presence of infinite-valence vertices.

We describe the Lipschitz metric and show how to compute it; we prove the existence of optimal maps; we describe geodesics represented by folding paths.

We show that train tracks representative of irreducible (hence hyperbolic) automorphisms exist and that their are metrically characterized as minimal displaced points, showing in particular that the set of train tracks is closed (in particular, answering to some questions raised in Axis in outer space (2011) concerning the axis bundle of irreducible automorphisms).

Finally, we include a proof of the existence of simplicial train tracks map without using Perron-Frobenius theory.

A direct corollary of this general viewpoint is an easy proof that relative train track maps exist in both the free group and free product case.