2018/07/08 by Stefano Francaviglia, Francaviglia, Stefano, Armando Martino +1
Mathematics · #20E06 #20E08 #20E36 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1807.02781
openalex publication_date 2018/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is the first of two papers in which we investigate the properties of the\ndisplacement functions of automorphisms of free groups (more generally, free\nproducts) on Culler-Vogtmann Outer space and its simplicial bordification - the\nfree splitting complex - with respect to the Lipschitz metric. The theory for\nirreducible automorphisms being well-developed, we concentrate on the reducible\ncase. Since we deal with the bordification, we develop all the needed tools in\nthe more general setting of deformation spaces, and their associated free\nsplitting complexes.\n In the present paper we study the local properties of the displacement\nfunction. In particular, we study its convexity properties and the behaviour at\nbordification points, by geometrically characterising its continuity-points. We\nprove that the global-simplex-displacement spectrum of Aut(Fn) is a\nwell-ordered subset of mathbb R, this being helpful for algorithmic\npurposes. We introduce a weaker notion of train tracks, which we call em\npartial train tracks (which coincides with the usual one for irreducible\nautomorphisms) and we prove that, for any automorphism, points of minimal\ndisplacement - minpoints - coincide with the marked metric graphs that support\npartial train tracks. We show that any automorphism, reducible or not, has a\npartial train track (hence a minpoint) either in the outer space or its\nbordification. We show that, given an automorphism, any of its invariant free\nfactors is seen in a partial train track map. In a subsequent paper we will\nprove that level sets of the displacement functions are connected, and we will\napply that result to solve certain decision problems.\n