Asymmetry of Outer space

By Yael Algom-Kfir, M. Bestvina. 2009

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We study the asymmetry of the Lipschitz metric d on Outer space. We introduce an (asymmetric) Finsler norm $${\|\cdot\|^L}$$ that induces d. There is an Out(Fn)-invariant “potential” Ψ defined on Outer space such that when $${\|\cdot\|^L}$$ is corrected by dΨ, the resulting norm is quasi-symmetric. As an application, we give new proofs of two theorems of Handel-Mosher, that d 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 $${f\in Out(F_n)}$$ and its inverse.

Subfields: Astronomy and Astrophysics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ff1ff8e04212617f51bf125501ac8e07fd69c108