Asymmetry of Outer Space

2009/10/28 by Yael Algom-Kfir, Algom-Kfir, Yael, Mladen Bestvina +1 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.0910.5408

openalex publication_date 2009/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the asymmetry of the Lipschitz metric d on Outer space. We introduce an (asymmetric) Finsler norm that induces d. There is an Out(F_n)-invariant potential Ψon Outer space such that when the Lipschitz norm is corrected by the derivative of Ψ, the resulting norm is quasisymmetric. As an application, we give new proofs of two theorems of Handel-Mosher, that the Lipschitz metric is quasi-symmetric when restricted to a thick part of Outer space, and that there is a uniform bound, depending only on the rank, on the ratio of logs of growth rates of any irreducible outer automorphism f in Out(F_n) and its inverse.

Citations

• Minimal stretch maps between hyperbolic surfaces

Cited by

• Relative twisting in Outer space
• Complete invariant geodesic metrics on outer spaces and Jacobian varieties of tropical curves

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