Asymmetry of Outer space

Yael Algom-Kfir, Mladen Bestvina

Research output: Contribution to journalArticlepeer-review


We study the asymmetry of the Lipschitz metric d on Outer space. We introduce an (asymmetric) Finsler norm {double pipe}·{double pipe}L that induces d. There is an Out(Fn)-invariant "potential" Ψ defined on Outer space such that when {double pipe}·{double pipe}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 ε Out(Fn) and its inverse.

Original languageEnglish
Pages (from-to)81-92
Number of pages12
JournalGeometriae Dedicata
Issue number1
StatePublished - Feb 2012
Externally publishedYes

Bibliographical note

Funding Information:
Acknowledgments We thank Bert Wiest and the referee for helpful comments. The second author gratefully acknowledges the support by the National Science Foundation


  • Automorphisms of the free group
  • Outer space

ASJC Scopus subject areas

  • Geometry and Topology


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