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

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 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.