[Paper Review] Metric properties of Outer Space

Stefano Francaviglia, Armando Martino|ePrints Soton (University of Southampton)|Mar 5, 2008

Geometric and Algebraic Topology5 references4 citations

TL;DR

This paper introduces and analyzes two metrics on Culler-Vogtmann Outer Space— a non-symmetric metric based on stretching factors and its symmetrized version—proving that folding paths are geodesics for the non-symmetric metric and quasi-geodesics for the symmetric metric when avoiding the thin part. The key contribution is establishing the symmetric metric's properness and showing that iterated automorphisms yield quasi-isometries, even for polynomial-growth cases.

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.

Motivation & Objective

• To develop a metric on Outer Space that mirrors the Teichmüller metric for surfaces, enabling geometric study of the outer automorphism group of a free group.
• To compare the non-symmetric stretching factor metric with its symmetrized version, analyzing their topological and geometric properties.
• To investigate the behavior of folding paths under these metrics, particularly their geodesic and quasi-geodesic properties.
• To examine the dynamics of automorphisms via their action on Outer Space, especially the quasi-isometric behavior of iterated orbits.
• To address open questions on coarse geometry, hyperbolicity, and the existence of geodesic axes for irreducible atoroidal automorphisms.

Proposed method

• Define a non-symmetric metric on Outer Space using the logarithm of the minimal Lipschitz constant (stretching factor) between marked metric graphs.
• Construct the symmetric metric by symmetrizing the non-symmetric metric, ensuring properness and compatibility with the length function topology.
• Use folding paths as canonical parameterized paths in Outer Space, constructed by iteratively folding edges in marked metric graphs.
• Apply Lemmas 7.7 and 7.8 to compute local speed and distance to target, showing the ratio of speed to distance is bounded below by 1/2.
• Establish quasi-geodesic behavior using the 4-point property and Lemma 7.2, proving uniform quasi-geodesic constants under bounded geometry.
• Analyze iterated automorphisms by tracking the folding path between a rose and its image under an iterate, showing it forms a quasi-geodesic with uniform constants.

Experimental results

Research questions

• RQ1 Are folding paths quasi-geodesics for the symmetric metric when they avoid the thin part of Outer Space?
• RQ2 Does the symmetric metric on Outer Space induce the same topology as the length function topology, and is it proper?
• RQ3 Can the orbit of a point under iteration of an automorphism be a quasi-isometry, and does this hold for polynomial-growth automorphisms?
• RQ4 Is there a geodesic axis in Outer Space for an irreducible atoroidal (iwip) automorphism?
• RQ5 Does the thick part of Outer Space exhibit hyperbolic-like properties, and what are the coarse geometric features such as flats or asymptotic cones?

Key findings

• The symmetric metric on Outer Space is proper: closed balls are compact, unlike the non-symmetric metric, which fails to be complete.
• Folding paths are geodesics for the non-symmetric metric, as established by the fact that stretching factors realize the distance at each point.
• Folding paths that avoid the thin part of Outer Space are (4,0)-quasi-geodesics for the symmetric metric, with uniform constants independent of the path.
• For any automorphism of exponential growth, the orbit map from ℤ to Outer Space is a quasi-isometry, confirming strong dynamical behavior.
• Even for polynomial-growth automorphisms, the folding path between a rose and its iterate is a (2,0)-quasi-geodesic, showing robustness of the result.
• The symmetric metric is not geodesic: a counterexample is provided, showing that not all pairs of points are connected by a geodesic path.