Envelopes in Outer Space

2019/07/15 by Christian Steinhart, Steinhart, Christian
Mathematics · #Geometric and Algebraic Topology #Advanced Combinatorial Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1907.06402

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 Isom(CV_n^red)=Isom(CV_n). As another implication we get that a geodesic ray in CV₂ becomes after a given length rigid.

Related

• Envelopes of the Thurston metric on Teichmüller space
• Geodesic Envelopes in the Thurston Metric on Teichmuller space
• Convexity of balls in the outer space
• Applications of Envelopes
• Asymptotic cones, bi-Lipschitz ultraflats, and the geometric rank of geodesics