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

This is the first of two papers in which we investigate the prop-erties of the displacement functions of automorphisms of free groups (moregenerally, free products) on Culler-Vogtmann Outer space and its simplicialbordification – the free splitting complex – with respect to the Lipschitz metric.The theory for irreducible automorphisms being well-developed, we concen-trate on the reducible case. Since we deal with the bordification, we developall the needed tools in the more general setting of deformation spaces, andtheir associated free splitting complexes.In the present paper we study the local properties of the displacementfunction. In particular, we study its convexity properties and the behaviourat bordification points, by geometrically characterising its continuity-points.We prove that the global-simplex-displacement spectrum ofAut(Fn) is a well-ordered subset ofR, this being helpful for algorithmic purposes. We introduce aweaker notion of train tracks, which we callpartial train tracks(which coincideswith the usual one for irreducible automorphisms) and we prove that, forany automorphism, points of minimal displacement – minpoints – coincidewith the marked metric graphs that support partial train tracks. We showthat any automorphism, reducible or not, has a partial train track (hence aminpoint) either in the outer space or its bordification. We show that, givenan automorphism, any of its invariant free factors is seen in a partial traintrack map.In a subsequent paper we will prove that level sets of the displacementfunctions are connected, and we will apply that result to solve certain decisionproblems.