Contracting orbits in Outer space

  • Spencer DowdallEmail author
  • Samuel J. Taylor


We show that strongly contracting geodesics in Outer space project to parameterized quasigeodesics in the free factor complex. This result provides a converse to a theorem of Bestvina–Feighn, and is used to give conditions for when a subgroup of \({{\,\mathrm{Out}\,}}(\mathbb {F})\) has a quasi-isometric orbit map into the free factor complex. It also allows one to construct many new examples of strongly contracting geodesics in Outer space.



The authors thank the referee for several helpful comments. The first named author was partially supported by NSF Grants DMS 1204814 and 1711089. The second named author was partially supported by NSF Grants DMS 1400498 and 1744551. We also acknowledge support from NSF Grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network) and from the GATSBY seminar at Brown and Yale.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA
  2. 2.Department of MathematicsTemple UniversityPhiladelphiaUSA

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