## 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

## Additional Information

**Radhika Gupta**- Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
- Address at time of publication: Department of Mathematics, Technion, Haifa 32000, Israel
- Email: radhikagup@technion.ac.il
- Received by editor(s): February 2, 2017
- Received by editor(s) in revised form: July 8, 2017
- Published electronically: October 18, 2017
- Additional Notes: The author was partially supported by the NSF grant of Mladen Bestvina (DMS-1607236)
- © Copyright 2017 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**21**(2017), 319-352 - MSC (2010): Primary 20F65
- DOI: https://doi.org/10.1090/ecgd/314
- MathSciNet review: 3713836