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Geodesic currents and length compactness for automorphisms of free groups

Author: Stefano Francaviglia
Journal: Trans. Amer. Math. Soc. 361 (2009), 161-176
MSC (2000): Primary 20F65
Published electronically: August 13, 2008
MathSciNet review: 2439402
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Abstract: We prove a compactness theorem for automorphisms of free groups. Namely, we show that the set of automorphisms keeping the length of the uniform current bounded is compact (up to conjugation). This implies that the spectrum of the length of the images of the uniform current is discrete, proving a conjecture of I. Kapovich.

References [Enhancements On Off] (What's this?)

  • [AFP00] Luigi Ambrosio, Nicola Fusco, and Diego Pallara.
    Functions of bounded variation and free discontinuity problems.
    Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292 (2003a:49002)
  • [Bon86] Francis Bonahon.
    Bouts des variétés hyperboliques de dimension $ 3$.
    Ann. of Math. (2), 124(1):71-158, 1986. MR 847953 (88c:57013)
  • [Coo87] Daryl Cooper.
    Automorphisms of free groups have finitely generated fixed point sets.
    J. Algebra, 111(2):453-456, 1987. MR 916179 (89a:20024)
  • [Fed69] Herbert Federer.
    Geometric measure theory.
    Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag, New York, Inc., New York, 1969. MR 0257325 (41:1976)
  • [Kap05] Ilya Kapovich.
    The frequency space of a free group.
    Internat. J. Alg. Comput. (special Gaeta Grigorchuk's 50's birthday issue), 15(5-6):939-969, 2005. MR 2197815 (2007a:20038)
  • [Kap06a] Ilya Kapovich.
    Clusters, currents and Whitehead's algorithm.
    Experimental Mathematics, 16(1):67-76, 2007. MR 2312978 (2008f:20043)
  • [Kap06b] Ilya Kapovich.
    Currents on free groups.
    In Topological and asymptotic aspects of group theory, volume 394 of Contemp. Math., pages 149-176.
    Amer. Math. Soc., Providence, RI, 2006. MR 2216713
  • [KKS05] Vadim Kaimanovich, Ilya Kapovich, and Paul Shupp.
    The subadditive ergodic theorem and generic stretching factors for free group automorphisms.
    Preprint, arXiv:math.GR/0504105, to appear in Israel J. Math., 2005.
  • [LS77] Roger C. Lyndon and Paul E. Shupp.
    Combinatorial group theory.
    Springer-Verlag, Berlin, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. MR 0577064 (58:28182)
  • [Mar95] R. Martin.
    Non-uniquely ergodic foliations of thin type, measured currents and automorphisms of free groups.
    Ph.D. thesis, University of California, Los Angeles, 1995.
  • [Whi36] J. H. C. Whitehead.
    On equivalent sets of elements in a free group.
    Ann. of Math. (2), 37(4):782-800, 1936. MR 1503309

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Additional Information

Stefano Francaviglia
Affiliation: Departament de Matemàtiques, Edifici C, Universitat Autònoma, 08193 Bellaterra (Barcelona) Spain

Keywords: Automorphisms, free groups, geodesic currents
Received by editor(s): May 22, 2006
Received by editor(s) in revised form: October 25, 2006
Published electronically: August 13, 2008
Additional Notes: The author was supported by a Marie Curie Intra European Fellowship.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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