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 [22] 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.

Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings

“…Once the above distance estimates have been established, the proof of Theorem 3.1 goes by checking Masur and Minsky's axioms [44,Theorem 2.3] for the set of φ-images in F S(G, F) of optimal liberal folding paths between simplicial trees with trivial edge stabilizers in O(G, F), which is coarsely transitive (the existence of optimal morphisms between splittings in O(G, F) follows from [15,Corollary 6.8], and there is a canonical way to build a folding path from a morphism, as recalled in Section 2.3).…”

“…We review work by Francaviglia and Martino [15]. Let G be a countable group and F be a free factor system of G. For all T, T ∈ O(G, F), we denote by Lip(T, T ) the infimal Lipschitz constant of an equivariant map from T to T .…”

Loxodromics for the cyclic splitting complex and their centralizers

“…We would like to remark that existence of an optimal morphism which is a train track map representing a relative fully irreducible outer automorphism is a special case of the results of [FM13] and [Mei15], for free products and deformation spaces, respectively. The authors of [FM13] develop metric theory for relative outer space for free products which is then used to show the existence of optimal maps. This requires considerable amount of work due to lack of applicability of Arzela-Ascoli theorem in this setting.…”

Asymmetry of outer space of a free product

“…Also, for convenience we normalise the length of edges and we suppose that the sum of the lengths of edges in E 1 (T ) is 1. Note that by a remark of [9], the hyperbolic elements of T ∈ O depends only on the space O and we denote them by Hyp(O). On the other hand, for a G-tree T as above, we can consider a lot of different metrics s.t.…”

“…There are a lot of analogies between the classical and the general Outer Space. Firstly, Francaviglia and Martino in [9] introduced and studied the Lipschitz metric for the general case. In the same paper, they proved as an application, the existence of train track representatives for (relative) IWIP automorphisms.…”