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Dynamics of Free Group Automorphisms

  • Peter Brinkmann
Part of the Trends in Mathematics book series (TM)

Abstract

We present a coarse convexity result for the dynamics of free group automorphisms: Given an automorphism ø of a finitely generated free group F, we show that for all xF and 0 ≤ iN, the length of ø i (x) is bounded above by a constant multiple of the sum of the lengths of x and ø N (x), with the constant depending only on ø.

Mathematics Subject Classification (2000)

37E30 

Keywords

Free group automorphisms train tracks 

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References

  1. [BF92]
    M. Bestvina and M. Feighn. A combination theorem for negatively curved groups. J. Differential Geom., 35(1):85–101, 1992.zbMATHMathSciNetGoogle Scholar
  2. [BFH97]
    M. Bestvina, M. Feighn, and M. Handel. Laminations, trees, and irreducible automorphisms of free groups. Geom. Funct. Anal., 7(2):215–244, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [BFH00]
    Mladen Bestvina, Mark Feighn, and Michael Handel. The Tits alternative for Out(F n). I. Dynamics of exponentially-growing automorphisms. Ann. of Math. (2), 151(2):517–623, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [BG]
    Martin R. Bridson and Daniel P. Groves. The quadratic isoperimetric inequality for mapping tori of free group automorphisms. arXiv:0802.1323.Google Scholar
  5. [BH92]
    Mladen Bestvina and Michael Handel. Train tracks and automorphisms of free groups. Ann. of Math. (2), 135(1):1–51, 1992.CrossRefMathSciNetGoogle Scholar
  6. [Bri]
    Peter Brinkmann. Detecting automorphic orbits in free groups. arXiv:0806.2889v1.Google Scholar
  7. [Bri00]
    Peter Brinkmann. Hyperbolic automorphisms of free groups. Geom. Funct. Anal., 10(5):1071–1089, 2000. arXiv:math.GR/9906008.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [Coo87]
    Daryl Cooper. Automorphisms of free groups have finitely generated fixed point sets. J. Algebra, 111(2):453–456, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [DV96]
    Warren Dicks and Enric Ventura. The group fixed by a family of injective endomorphisms of a free group. American Mathematical Society, Providence, RI, 1996.zbMATHGoogle Scholar
  10. [Gan59]
    F.R. Gantmacher. The theory of matrices. Vols. 1, 2. Chelsea Publishing Co., New York, 1959. Translated by K.A. Hirsch.Google Scholar
  11. [Ger94]
    S.M. Gersten. The automorphism group of a free group is not a CAT(0) group. Proc. Amer. Math. Soc., 121(4):999–1002, 1994.zbMATHMathSciNetGoogle Scholar
  12. [Mac00]
    N. Macura. Quadratic isoperimetric inequality for mapping tori of polynomially growing automorphisms of free groups. Geom. Funct. Anal., 10(4):874–901, 2000.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Peter Brinkmann
    • 1
  1. 1.Department of MathematicsThe City College of CUNYNew YorkUSA

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