Envelopes in Outer Space

Abstract

We study the geometry of Outer Space $CV_n$ in regard of the asymmetric Lipschitz metric via envelopes, that is the set of all geodesics between two points. In the simplicial structure of $CV_n$ the envelopes are polytopes. We construct a piecewise unique geodesic between any two points in $CV_n$ by concatenating edges of these polytopes. In fact rigid geodesics can be identified with edges of out- and in-envelopes, that is the set of all geodesics from or to a base point with a given maximally stretched path. We introduce a notion of general position for pairs of points which is a dense and open condition. Using this we will show, that for almost all pairs of points in $CV_n$ their envelopes have dimension $3n-4$. Whenever an envelope passes a face, it might change its dimension. This determines the simplicial structure of reduced Outer Space via the Lipschitz metric which implies $\mathrm{Isom}(CV_n^{red})=\mathrm{Isom}(CV_n)$. As another implication we get that a geodesic ray in $CV_2$ becomes after a given length rigid.