Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks

TL;DR

This paper investigates the properties of displacement functions of automorphisms of free groups on Outer space, focusing on reducible automorphisms, and introduces partial train tracks to characterize minimal displacement points.

Contribution

It develops the theory of displacement functions for reducible automorphisms, introduces partial train tracks, and characterizes minpoints in the context of Outer space and its bordification.

Findings

Displacement spectrum of Aut(F_n) is well-ordered.

Partial train tracks coincide with minpoints for automorphisms.

Every automorphism has a partial train track or minpoint in Outer space or its bordification.

Abstract

This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free splitting complex - with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of $Aut(F_n)$ is a well-ordered subset of $\mathbb R$, this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call {\em partial train tracks} (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement - minpoints - coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems.