Geometric and Functional Analysis

, Volume 19, Issue 5, pp 1426–1467 | Cite as

Intersection Form, Laminations and Currents on Free Groups

Open Access


Let F be a free group of rank N ≥ 2, let μ be a geodesic current on F and let T be an \({\mathbb{R}}\)-tree with a very small isometric action of F. We prove that the geometric intersection number \({{\langle T,\mu \rangle}}\) is equal to zero if and only if the support of μ is contained in the dual algebraic lamination L2(T) of T. Applying this result, we obtain a generalization of a theorem of Francaviglia regarding length spectrum compactness for currents with full support. We use the main result to obtain “unique ergodicity” type properties for the attracting and repelling fixed points of atoroidal iwip elements of Out(F) when acting both on the compactified outer Space and on the projectivized space of currents. We also show that the sum of the translation length functions of any two “sufficiently transverse” very small F-trees is bilipschitz equivalent to the translation length function of an interior point of the outer space. As another application, we define the notion of a filling element in F and prove that filling elements are “nearly generic” in F. We also apply our results to the notion of bounded translation equivalence in free groups.

Keywords and phrases

Free groups Outer space geodesic currents 

2000 Mathematics Subject Classification

Primary 20F Secondary 57M 37B 37D 


  1. B.
    Bass H. (1993) Covering theory for graphs of groups. J. Pure Appl. Algebra 89(1-2): 3–47MATHCrossRefMathSciNetGoogle Scholar
  2. BeF1.
    Bestvina M., Feighn M. (2000) The topology at infinity of Out(F n). Invent. Math. 140(3): 651–692MATHCrossRefMathSciNetGoogle Scholar
  3. BeF2.
    M. Bestvina, M. Feighn, A hyperbolic Out(F n) complex, preprint (2008); arXiv(0808.3730Google Scholar
  4. BeF3.
    M. Bestvina, M. Feighn, Outer limits, preprint (1993);
  5. BeFH1.
    Bestvina M., Feighn M., Handel M. (1997) Laminations, trees, and irreducible automorphisms of free groups. Geom. Funct. Anal. 7(2): 215–244MATHCrossRefMathSciNetGoogle Scholar
  6. BeFH2.
    Bestvina M., Feighn M., Handel M. (2000) The Tits alternative for Out(F n). I. Dynamics of exponentially-growing automorphisms. Ann. of Math. (2) 151(2): 517–623MATHCrossRefMathSciNetGoogle Scholar
  7. BeFH3.
    Bestvina M., Feighn M., Handel M. (2005) The Tits alternative for Out(F n). II. A Kolchin type theorem. Ann. of Math. (2) 161(1): 1–59MATHCrossRefMathSciNetGoogle Scholar
  8. BeH.
    Bestvina M., Handel M. (1992) Train tracks and automorphisms of free groups. Ann. of Math. (2) 135(1): 1–51CrossRefMathSciNetGoogle Scholar
  9. Bo1.
    Bonahon F. (1986) Bouts des variétés hyperboliques de dimension 3. Ann. of Math. (2) 124(1): 71–158CrossRefMathSciNetGoogle Scholar
  10. Bo2.
    Bonahon F. (1988) The geometry of Teichmüller space via geodesic currents. Invent. Math. 92(1): 139–162MATHCrossRefMathSciNetGoogle Scholar
  11. Bo3.
    F. Bonahon, Geodesic currents on negatively curved groups, Arboreal Group Theory (Berkeley, CA, 1988), Math. Sci. Res. Inst. Publ., 19, Springer, New York (1991), 143–168.Google Scholar
  12. CF.
    D. Calegari, K. Fujiwara, Combable functions, quasimorphisms, and the central limit theorem, preprint;
  13. CoL.
    Cohen M., Lustig M. (1995) Very small group actions on R-trees and Dehn twist automorphisms. Topology 34(3): 575–617MATHCrossRefMathSciNetGoogle Scholar
  14. Coo.
    Cooper D. (1987) Automorphisms of free groups have finitely generated fixed point sets. J. Algebra 111(2): 453–456MATHCrossRefMathSciNetGoogle Scholar
  15. CouHL1.
    Coulbois T., Hilion A., Lustig M. (2008) \({\mathbb{R}}\) -trees and laminations for free groups I: Algebraic laminations. J. Lond. Math. Soc. (2) 78(3): 723–736MATHCrossRefMathSciNetGoogle Scholar
  16. CouHL2.
    Coulbois T., Hilion A., Lustig M. (2008) \({\mathbb{R}}\) -trees and laminations for free groups II: The dual lamination of an \({\mathbb{R}}\) -tree. J. Lond. Math. Soc. (2) 78(3): 737–754MATHCrossRefMathSciNetGoogle Scholar
  17. CouHL3.
    Coulbois T., Hilion A., Lustig M. (2008) \({\mathbb{R}}\) -trees and laminations for free groups III: Currents and dual \({\mathbb{R}}\) -tree metrics. J. Lond. Math. Soc. 78(3): 755–766MATHCrossRefMathSciNetGoogle Scholar
  18. CuV.
    Culler M., Vogtmann K. (1986) Moduli of graphs and automorphisms of free groups. Invent. Math. 84(1): 91–119MATHCrossRefMathSciNetGoogle Scholar
  19. GJLL.
    Gaboriau D., Jaeger A., Levitt G., Lustig M. (1998) An index for counting fixed points of automorphisms of free groups. Duke Math. J. 93(3): 425–452MATHCrossRefMathSciNetGoogle Scholar
  20. GyH.
    E. Ghys, P. de la Harpe (eds.), Sur les groupes hyperboliques d’aprés Mikhail Gromov, Birkhäuser, Progress in Mathematics 83 (1990).Google Scholar
  21. Gu.
    Guirardel V. (1998) Approximations of stable actions on R-trees. Comment. Math. Helv. 73(1): 89–121MATHCrossRefMathSciNetGoogle Scholar
  22. F.
    Francaviglia S. (2009) Geodesic currents and length compactness for automorphisms of free groups. Trans. Amer. Math. Soc. 361(1): 161–176MATHCrossRefMathSciNetGoogle Scholar
  23. H.
    U. Hamenstädt, subgroups of Out(F n), a talk at the workshop “Discrete Groups and Geometric Structures”, Kortrijk, Belgium, May 2008.Google Scholar
  24. KKS.
    Kaimanovich V., Kapovich I., Schupp P. (2007) The subadditive ergodic theorem and generic stretching factors for free group automorphisms. Israel J. Math. 157: 1–46MATHCrossRefMathSciNetGoogle Scholar
  25. Ka1.
    Kapovich I. (1997) Quasiconvexity and amalgams. Int. J. Alg. Comput. 7(6): 771–811MATHCrossRefMathSciNetGoogle Scholar
  26. Ka2.
    Kapovich I. (2001) The combination theorem and quasiconvexity. Intern. J. Alg. Comput. 11(2): 185–216MATHCrossRefMathSciNetGoogle Scholar
  27. Ka3.
    Kapovich I. (2005) The frequency space of a free group. Internat. J. Alg. Comput. 15(5-6): 939–969MATHCrossRefMathSciNetGoogle Scholar
  28. Ka4.
    I. Kapovich, Currents on free groups, Topological and Asymptotic Aspects of Group Theory (R. Grigorchuk, M. Mihalik, M. Sapir, Z. Sunik, eds.), AMS Contemporary Mathematics Series, 394 (2006), 149–176.Google Scholar
  29. Ka5.
    Kapovich I. (2007) Clusters, currents and Whitehead’s algorithm. Experimental Mathematics 16(1): 67–76MATHMathSciNetGoogle Scholar
  30. KaLSS.
    Kapovich I., Levitt G., Schupp P., Shpilrain V. (2007) Translation equivalence in free groups. Transact. Amer. Math. Soc. 359(4): 1527–1546MATHCrossRefMathSciNetGoogle Scholar
  31. KaLu1.
    Kapovich I., Lustig M. (2007) The actions of Out(F k) on the boundary of outer space and on the space of currents: minimal sets and equivariant incompatibility. Ergodic Theory Dynam. Systems 27(3): 827–847MATHCrossRefMathSciNetGoogle Scholar
  32. KaLu2.
    Kapovich I., Lustig M. (2009) Geometric intersection number and analogues of the curve complex for free groups. Geom. Topol. 13: 1805–1833MATHCrossRefMathSciNetGoogle Scholar
  33. KaM.
    Kapovich I., Myasnikov A. (2002) Stallings foldings and the subgroup structure of free groups. J. Algebra 248(2): 608–668MATHCrossRefMathSciNetGoogle Scholar
  34. KaN.
    Kapovich I., Nagnibeda T. (2007) The Patterson–Sullivan embedding and minimal volume entropy for outer space. Geom. Funct. Anal. 17(4): 1201–1236MATHCrossRefMathSciNetGoogle Scholar
  35. Kap.
    M. Kapovich, Hyperbolic Manifolds and Discrete Groups, Birkhäuser, 2001.Google Scholar
  36. L.
    Leininger C.J. (2003) Equivalent curves in surfaces. Geom. Dedicata 102: 151–177MATHCrossRefMathSciNetGoogle Scholar
  37. Le.
    Lee D. (2006) Translation equivalent elements in free groups. J. Group Theory 9(6): 809–814MATHCrossRefMathSciNetGoogle Scholar
  38. LevL.
    Levitt G., Lustig M. (2003) Irreducible automorphisms of F n have north-south dynamics on compactified outer space. J. Inst. Math. Jussieu 2(1): 59–72MATHCrossRefMathSciNetGoogle Scholar
  39. Lu.
    M. Lustig, A generalized intersection form for free groups, preprint (2004).Google Scholar
  40. M.
    R. Martin, Non-Uniquely Ergodic Foliations of Thin Type, Measured Currents and Automorphisms of Free Groups, PhD Thesis, 1995.Google Scholar
  41. P.
    Paulin F. (1989) The Gromov topology on R-trees. Topology Appl. 32(3): 197–221MATHCrossRefMathSciNetGoogle Scholar
  42. S.
    Serre J.-P. (1980) Trees. Springer-Verlag, Berlin-New YorkMATHGoogle Scholar
  43. Sh.
    Sharp R. (2010) Distortion and entropy for automorphisms of free groups. Discr. Contin. Dynam. Syst. 26(1): 347–363CrossRefGoogle Scholar
  44. Sk.
    R. Skora, Deformations of length functions in groups, preprint, Columbia University (1989).Google Scholar
  45. St.
    M. Steiner, Gluing Data and Group Actions on \({\mathbb{R}}\) -Trees, Thesis, Columbia University (1988).Google Scholar
  46. V.
    Vogtmann K. (2002) Automorphisms of free groups and outer space. Geometriae Dedicata 94: 1–31MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Mathématiques (LATP)Université Paul Cézanne -Aix Marseille IIIMarseille 20France

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