Metric properties of Outer Space

TL;DR

This paper introduces and analyzes new metric structures on Culler-Vogtmann space, exploring their properties, computational aspects, and behavior under automorphisms, with implications for understanding the geometry of Outer space.

Contribution

It defines and compares symmetric and non-symmetric metrics on Outer space, providing methods for computing stretching factors and analyzing geodesic properties of folding paths.

Findings

Metrics are related, with one being a symmetrized version of the other.

Folding paths are geodesic for the non-symmetric metric.

Folding paths are quasi-geodesic when outside the thin part of Outer space.

Abstract

We define metrics on Culler-Vogtmann space, which are an analogue of the Teichmuller metric and are constructed using stretching factors. In fact the metrics we study are related, one being a symmetrised version of the other. We investigate the basic properties of these metrics, showing the advantages and pathologies of both choices. We show how to compute stretching factors between marked metric graphs in an easy way and we discuss the behaviour of stretching factors under iterations of automorphisms. We study metric properties of folding paths, showing that they are geodesic for the non-symmetric metric and, if they do not enter the thin part of Outer space, quasi-geodesic for the symmetric metric.