The metric completion of Outer Space

Research output: Contribution to journalArticlepeer-review


We develop the theory of a metric completion of an asymmetric metric space. We characterize the points on the boundary of Outer Space that are in the metric completion of Outer Space with the Lipschitz metric. We prove that the simplicial completion, the subset of the completion consisting of simplicial tree actions, is homeomorphic to the free splitting complex. We use this to give a new proof of a theorem by Francaviglia and Martino that the isometry group of Outer Space is isomorphic to Out (Fn) for n≥ 3 and to PSL (2 , Z) for n= 2.

Original languageEnglish
Pages (from-to)191-230
Number of pages40
JournalGeometriae Dedicata
Issue number1
StatePublished - 1 Feb 2020

Bibliographical note

Publisher Copyright:
© 2019, Springer Nature B.V.


  • Asymmetric metric space
  • Metric completion
  • Out(F)
  • Outer Space
  • Quasimetric space

ASJC Scopus subject areas

  • Geometry and Topology


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