# Envelopes in Outer Space

## TL;DR

This paper explores the geometric structure of Outer Space using the Lipschitz metric, revealing how envelopes form polytopes and how geodesics can be uniquely constructed, with implications for the space's isometry group.

## Contribution

It introduces a novel geometric framework for Outer Space using envelopes, characterizes geodesics, and determines the space's isometry group under the Lipschitz metric.

## Key findings

- Envelopes are polytopes in the simplicial structure of $CV_n$.
- Almost all pairs of points have envelopes of dimension $3n-4$.
- The isometry group of reduced Outer Space equals that of 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.