Asymmetry of Outer Space

Details Asymmetry of Outer Space Asymmetry of Outer Space

2009-10-28

arxiv.org/abs/0910.5408v3

Mathematics

Group Theory

15 pages, accepted to Geometriae Dedicata, omitted the comment about the potential function in rank 2 being equal to injrad (b

Scientific paper

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.

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