Geometriae Dedicata

, Volume 166, Issue 1, pp 307–348 | Cite as

Subset currents on free groups

Original Paper

Abstract

We introduce and study the space \({{\mathcal{S}{\rm Curr} (F_N)}}\) of subset currents on the free group FN, and, more generally, on a word-hyperbolic group. A subset current on FN is a positive FN-invariant locally finite Borel measure on the space \({{\mathfrak{C}_N}}\) of all closed subsets of ∂FN consisting of at least two points. The well-studied space Curr(FN) of geodesics currents–positive FN-invariant locally finite Borel measures defined on pairs of different boundary points–is contained in the space of subset currents as a closed \({{\mathbb{R}}}\)-linear Out(FN)-invariant subspace. Much of the theory of Curr(FN) naturally extends to the \({{\mathcal{S}\;{\rm Curr} (F_N)}}\) context, but new dynamical, geometric and algebraic features also arise there. While geodesic currents generalize conjugacy classes of nontrivial group elements, a subset current is a measure-theoretic generalization of the conjugacy class of a nontrivial finitely generated subgroup in FN. If a free basis A is fixed in FN, subset currents may be viewed as FN-invariant measures on a “branching” analog of the geodesic flow space for FN, whose elements are infinite subtrees (rather than just geodesic lines) of the Cayley graph of FN with respect to A. Similarly to the case of geodesics currents, there is a continuous Out(FN)-invariant “co-volume form” between the Outer space cvN and the space \({{\mathcal{S}\;{\rm Curr} (F_N)}}\) of subset currents. Given a tree \({{T \in {\rm cv}_N}}\) and the “counting current” \({{\eta_H \in \mathcal{S}\;{\rm Curr} (F_N)}}\) corresponding to a finitely generated nontrivial subgroup H ≤  FN, the value \({{\langle T, \eta_H \rangle}}\) of this intersection form turns out to be equal to the co-volume of H, that is the volume of the metric graph TH/H, where \({{T_H \subseteq T}}\) is the unique minimal H-invariant subtree of T. However, unlike in the case of geodesic currents, the co-volume form \({{{\rm cv}_N \times \mathcal{S}\;{\rm Curr}(F_N)\; \to [0,\infty)}}\) does not extend to a continuous map \({{\overline{{\rm cv}}_N \times \mathcal{S}\; {\rm Curr} (F_N) \to [0,\infty)}}\).

Keywords

Free groups Geodesic currents Outer space Automorphisms of free groups 

Mathematics Subject Classification (2000)

Primary 20F Secondary 57M 37B 37D 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abert, M., Glasner, Y., Virag, B.: Kesten’s theorem for invariant random subgroups, preprint. arXiv:1201.3399 (2012)Google Scholar
  2. 2.
    Arnoux P., Berthé V., Fernique T., Jamet D.: Functional stepped surfaces, flips, and generalized substitutions. Theor. Comput. Sci. 380(3), 251–265 (2007)CrossRefMATHGoogle Scholar
  3. 3.
    Bassino F., Nicaud C., Weil P.:: Random generation of finitely generated subgroups of a free group. Int. J. Algebra Comput. 18(2), 375–405 (2008)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Benjamini I., Schramm O.: Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6(23), 1–23 (2001)MathSciNetGoogle Scholar
  5. 5.
    Bestvina, M., Feighn, M.: Outer Limits, preprint. http://andromeda.rutgers.edu/~feighn/papers/outer.pdf (1993)
  6. 6.
    Bestvina M., Feighn M., Handel M.: Laminations, trees, and irreducible automorphisms of free groups. Geom. Funct. Anal. 7(2), 215–244 (1997)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bestvina M., Feighn M.: A hyperbolic Out(F n) complex. Groups Geom. Dyn. 4(1), 31–58 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bonahon F.: Bouts des variétés hyperboliques de dimension 3. Ann. of Math.(2) 124(1), 71–158 (1986)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bonahon F.: The geometry of Teichmüller space via geodesic currents. Invent. Math. 92(1), 139–162 (1988)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bowen L.: Periodicity and circle packings of the hyperbolic plane. Geom. Dedicata 102, 213–236 (2003)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bowen L.: Free groups in lattices. Geom. Topol. 13(5), 3021–3054 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bowen, L.: Random walks on coset spaces with applications to Furstenberg entropy, preprint. arXiv:1008.4933 (2010)Google Scholar
  13. 13.
    Bowen, L.: Invariant random subgroups of the free group, preprint. arXiv:1204.5939 (2012)Google Scholar
  14. 14.
    Carette, M., Francaviglia, S., Kapovich, I., Martino, A.: Spectral rigidity of automorphic orbits in free groups. Alg. Geom. Topol. 12, 1457–1486 (2012)Google Scholar
  15. 15.
    Cohen M., Lustig M.: Very small group actions on R-trees and Dehn twist automorphisms. Topology 34(3), 575–617 (1995)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Clay M., Pettet A.: Currents twisting and nonsingular matrices. Commentarii Mathematici Helvetici 87(2), 384–407 (2012)MathSciNetGoogle Scholar
  17. 17.
    Coulbois T., Hilion A., Lustig M.: \({{\mathbb{R}}}\)-trees and laminations for free groups I: Algebraic laminations. J. Lond. Math. Soc. (2) 78(3), 723–736 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Coulbois T., Hilion A., Lustig M.: \({{\mathbb{R}}}\)-trees and laminations for free groups II: the dual lamination of an \({{\mathbb{R}}}\)-tree. J. Lond. Math. Soc. (2) 78(3), 737–754 (2008)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Coulbois T., Hilion A., Lustig M.: \({{\mathbb{R}}}\)-trees and laminations for free groups III: Currents and dual \({{\mathbb{R}}}\) –tree metrics. J. Lond. Math. Soc. (2) 78(3), 755–766 (2008)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Culler M., Vogtmann K.: Moduli of graphs and automorphisms of free groups. Invent. Math. 84(1), 91–119 (1986)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    D’Angeli D., Donno A., Matter M., Nagnibeda T.: Schreier graphs of the Basilica group. J. Mod. Dyn. 4(1), 167–205 (2010)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Dani S.G.: On conjugacy classes of closed subgroups and stabilizers of Borel actions of Lie groups. Ergod. Theory Dyn. Syst. 22(6), 1697–1714 (2002)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Elek G.: On the limit of large girth graph sequences. Combinatorica 30(5), 553–563 (2010)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Francaviglia S.: Geodesic currents and length compactness for automorphisms of free groups. Trans. Am. Math. Soc. 361(1), 161–176 (2009)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Grigorchuk R.: Some topics of dynamics of group actions on rooted trees. Proc. Steklov Inst. Math. 273, 64–175 (2011)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Grigorchuk R., Kaimanovich V.A., Nagnibeda T.: Ergodic properties of boundary actions and Nielsen–Schreier theory. Adv. Math. 230(3), 1340–1380 (2012)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Guirardel V.: Dynamics of Out (F n) on the boundary of outer space. Ann. Sci. École Norm. Sup. (4) 33(4), 433–465 (2000)MathSciNetMATHGoogle Scholar
  28. 28.
    Hamenstädt, U.: Lines of minima in outer space. November 2009, preprint. arXiv:0911.3620 (2009)Google Scholar
  29. 29.
    Hart K.P., Nagata J.-I., Vaughan J.E. (ed.): Encyclopedia of General Topology. Elsevier, Amsterdam (2004)Google Scholar
  30. 30.
    Kapovich, I.: Currents on free groups. In: Grigorchuk, R., Mihalik, M., Sapir, M., Sunik, Z. (eds.) Topological and Asymptotic Aspects of Group Theory, AMS Contemporary Mathematics Series, vol. 394, pp. 149–176 (2006)Google Scholar
  31. 31.
    Kapovich I.: Clusters, currents and Whitehead’s algorithm. Exp. Math. 16(1), 67–76 (2007)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Kapovich I.: Random length-spectrum rigidity for free groups. Proc. AMS 140(5), 1549–1560 (2012)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Kapovich I., Levitt G., Schupp P., Shpilrain V.: Translation equivalence in free groups. Trans. Am. Math. Soc. 359(4), 1527–1546 (2007)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Kapovich I., Lustig M.: The actions of Out(F k) on the boundary of outer space and on the space of currents: minimal sets and equivariant incompatibility. Ergod. Theory Dyn. Syst. 27(3), 827–847 (2007)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Kapovich I., Lustig M.: Geometric intersection number and analogues of the curve complex for free groups. Geom. Topol. 13, 1805–1833 (2009)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Kapovich I., Lustig M.: Intersection form, laminations and currents on free groups. Geom. Funct. Anal. (GAFA) 19(5), 1426–1467 (2010)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Kapovich, I., Lustig, M.: Domains of proper dicontinuity on the boundary of outer space. Ill. J. Math. 54(1):89–108. Special issue dedicated to Paul Schupp (2010)Google Scholar
  38. 38.
    Kapovich I., Lustig M.: Ping-pong and outer space. J. Topol. Anal. 2(2), 173–201 (2010)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Kapovich I., Myasnikov A.: Stallings foldings and the subgroup structure of free groups. J. Algebra 248(2), 608–668 (2002)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Kapovich I., Nagnibeda T.: The Patterson–Sullivan embedding and minimal volume entropy for outer space. Geom. Funct. Anal. (GAFA) 17(4), 1201–1236 (2007)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Kapovich, I., Nagnibeda, T.: Geometric Entropy of Geodesic Currents on Free Groups. Dynamical Numbers: Interplay Between Dynamical Systems and Number Theory, Contemporary Mathematics Series. American Mathematical Society, Providence, RI, pp. 149–176 (2010)Google Scholar
  42. 42.
    Kapovich I., Short H.: Greenberg’s theorem for quasiconvex subgroups of word hyperbolic groups. Can. J. Math. 48(6), 1224–1244 (1996)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Lee D.: Translation equivalent elements in free groups. J. Group Theory 9(6), 809–814 (2006)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Lee D.: An algorithm that decides translation equivalence in a free group of rank two. J. Group Theory 10(4), 561–569 (2007)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Lee D., Ventura E.: Volume equivalence of subgroups of free groups. J. Algebra 324(2), 195–217 (2010)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Levitt G., Lustig M.: Irreducible automorphisms of F n have north–south dynamics on compactified outer space. J. Inst. Math. Jussieu 2(1), 59–72 (2003)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Martin, R.: Non-uniquely ergodic foliations of thin type, measured currents and automorphisms of free groups. Ph.D. Thesis (1995)Google Scholar
  48. 48.
    Mineyev I.: Submultiplicativity and the Hanna Neumann conjecture. Ann. Math. (2) 175(1), 393–414 (2012)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Savchuk, D.: Schreier Graphs of actions of Thompsons Group F on the unit interval and on the cantor set, preprint. arXiv:1105.4017 (2011)Google Scholar
  50. 50.
    Stallings J.: Topology of finite graphs. Invent. Math. 71(3), 551–565 (1983)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Stuck G., Zimmer R.J.: Stabilizers for ergodic actions of higher rank semisimple groups. Ann. Math. (2) 139(3), 723–747 (1994)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Vershik, A.: Nonfree actions of countable groups and their characters. arXiv:1012.4604Google Scholar
  53. 53.
    Vershik, A.: Totally nonfree actions and infinite symmetric group, preprint. arXiv:1109.3413 (2011)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Section de mathématiquesUniversité de GenèveGenevaSwitzerland

Personalised recommendations