Asymmetry of Outer Space

TL;DR

This paper investigates the asymmetry of the Lipschitz metric in Outer space, introducing an invariant potential to achieve quasisymmetry and providing new proofs for key properties of the metric related to automorphisms.

Contribution

It introduces an Out(F_n)-invariant potential that corrects the Lipschitz norm, leading to quasisymmetry and new proofs of existing theorems on Outer space metrics.

Findings

The Lipschitz metric becomes quasisymmetric after correction.

New proofs for the quasi-symmetry of the metric on thick parts.

Bound on the ratio of growth rates for automorphisms and their inverses.

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 \Psi on Outer space such that when the Lipschitz norm is corrected by the derivative of \Psi, 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.