Relative currents

Gupta, Radhika

doi:10.1090/ecgd/314

Relative currents
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by Radhika Gupta PDF

Conform. Geom. Dyn. 21 (2017), 319-352 Request permission

Abstract:

In this paper we define currents relative to a free factor system. We prove that a fully irreducible outer automorphism relative to a free factor system acts with uniform north-south dynamics on a subspace of the space of projective relative currents.

References

Mladen Bestvina and Mark Feighn, Hyperbolicity of the complex of free factors, Adv. Math. 256 (2014), 104–155. MR 3177291, DOI 10.1016/j.aim.2014.02.001
M. Bestvina, M. Feighn, and M. Handel, Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7 (1997), no. 2, 215–244. MR 1445386, DOI 10.1007/PL00001618
Mladen Bestvina, Mark Feighn, and Michael Handel, The Tits alternative for $\textrm {Out}(F_n)$. I. Dynamics of exponentially-growing automorphisms, Ann. of Math. (2) 151 (2000), no. 2, 517–623. MR 1765705, DOI 10.2307/121043
Mladen Bestvina and Michael Handel, Train tracks and automorphisms of free groups, Ann. of Math. (2) 135 (1992), no. 1, 1–51. MR 1147956, DOI 10.2307/2946562
Francis Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), no. 1, 139–162. MR 931208, DOI 10.1007/BF01393996
N. Bourbaki. Éléments de mathématique. Fasc. XIII. Livre VI: Intégration. Chapitres 1, 2, 3 et 4: Inégalités de convexité, Espaces de Riesz, Mesures sur les espaces localement compacts, Prolongement d’une mesure, Espaces $L^{p}$. Deuxième édition revue et augmentée. Actualités Scientifiques et Industrielles, No. 1175. Hermann, Paris, 1965.
P. Brinkmann, Hyperbolic automorphisms of free groups, Geom. Funct. Anal. 10 (2000), no. 5, 1071–1089. MR 1800064, DOI 10.1007/PL00001647
Daryl Cooper, Automorphisms of free groups have finitely generated fixed point sets, J. Algebra 111 (1987), no. 2, 453–456. MR 916179, DOI 10.1016/0021-8693(87)90229-8
Mark Feighn and Michael Handel, The recognition theorem for $\textrm {Out}(F_n)$, Groups Geom. Dyn. 5 (2011), no. 1, 39–106. MR 2763779, DOI 10.4171/GGD/116
R. Gupta, Loxodromic elements for the relative free factor complex, arXiv:1612.04261, December 2016.
M. Handel and L. Mosher, Subgroup decomposition in Out($F_n$): Introduction and Research Announcement, arXiv:1302.2681, February 2013.
M. Handel and L. Mosher, Relative free splitting and relative free factor complexes I: Hyperbolicity, arXiv:1407.3508v1, 2014.
Ilya Kapovich, The frequency space of a free group, Internat. J. Algebra Comput. 15 (2005), no. 5-6, 939–969. MR 2197815, DOI 10.1142/S0218196705002700
Ilya Kapovich, Currents on free groups, Topological and asymptotic aspects of group theory, Contemp. Math., vol. 394, Amer. Math. Soc., Providence, RI, 2006, pp. 149–176. MR 2216713, DOI 10.1090/conm/394/07441
Ilya Kapovich and Martin Lustig, Geometric intersection number and analogues of the curve complex for free groups, Geom. Topol. 13 (2009), no. 3, 1805–1833. MR 2496058, DOI 10.2140/gt.2009.13.1805
M. Lustig and C. Uyanik. North-South dynamics of hyperbolic free group automorphisms on the space of currents, ArXiv e-prints, September 2015.
Reiner Martin, Non-uniquely ergodic foliations of thin-type, measured currents and automorphisms of free groups, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)–University of California, Los Angeles. MR 2693216
Howard A. Masur and Yair N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103–149. MR 1714338, DOI 10.1007/s002220050343
Martine Queffélec, Substitution dynamical systems—spectral analysis, Lecture Notes in Mathematics, vol. 1294, Springer-Verlag, Berlin, 1987. MR 924156, DOI 10.1007/BFb0081890
John R. Stallings, Whitehead graphs on handlebodies, Geometric group theory down under (Canberra, 1996) de Gruyter, Berlin, 1999, pp. 317–330. MR 1714852
Caglar Uyanik, Dynamics of hyperbolic iwips, Conform. Geom. Dyn. 18 (2014), 192–216. MR 3273532, DOI 10.1090/S1088-4173-2014-00270-7
Caglar Uyanik, Generalized north-south dynamics on the space of geodesic currents, Geom. Dedicata 177 (2015), 129–148. MR 3370027, DOI 10.1007/s10711-014-9982-2
J. H. C. Whitehead, On Certain Sets of Elements in a Free Group, Proc. London Math. Soc. (2) 41 (1936), no. 1, 48–56. MR 1575455, DOI 10.1112/plms/s2-41.1.48