The free splitting complex of a free group, II: Loxodromic outer automorphisms
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- by Michael Handel and Lee Mosher PDF
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Abstract:
We study the loxodromic elements for the action of $\mathsf {Out}(F_n)$ on the free splitting complex of the rank $n$ free group $F_n$. Each outer automorphism is either loxodromic or has bounded orbits without any periodic point, or has a periodic point; all three possibilities can occur. Two loxodromic elements are either coaxial or independent, meaning that their attracting-repelling fixed point pairs on the Gromov boundary of the free splitting complex are either equal or disjoint as sets. Each alternative is characterized in terms of attracting laminations; in particular, an outer automorphism is loxodromic if and only if it has a filling attracting lamination. As an application, each attracting lamination determines its corresponding repelling lamination independent of the outer automorphism. As part of this study, we describe the structure of the subgroup of $\mathsf {Out}(F_n)$ that stabilizes the fixed point pair of a given loxodromic outer automorphism, and we give examples which show that this subgroup need not be virtually cyclic. As an application, the action of $\mathsf {Out}(F_n)$ on the free splitting complex is not acylindrical, and its loxodromic elements do not all satisfy the weak proper discontinuity (WPD) property of Bestvina and Fujiwara.References
- Yael Algom-Kfir, Strongly contracting geodesics in outer space, Geom. Topol. 15 (2011), no. 4, 2181–2233. MR 2862155, DOI 10.2140/gt.2011.15.2181
- Yael Algom-Kfir and Catherine Pfaff, Normalizers and centralizers of cyclic subgroups generated by lone axis fully irreducible outer automorphisms, New York J. Math. 23 (2017), 365–381. MR 3649663
- Mladen Bestvina, Ken Bromberg, and Koji Fujiwara, Constructing group actions on quasi-trees and applications to mapping class groups, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 1–64. MR 3415065, DOI 10.1007/s10240-014-0067-4
- Mladen Bestvina and Koji Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002), 69–89. MR 1914565, DOI 10.2140/gt.2002.6.69
- 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
- M. Bestvina, M. Feighn, and M. Handel, The Tits alternative for $\textrm {Out}(F_ n)$. I. Dynamics of exponentially-growing automorphisms., Ann. of Math. 151 (2000), no. 2, 517–623.
- Mladen Bestvina, Mark Feighn, and Michael Handel, Solvable subgroups of $\textrm {Out}(F_n)$ are virtually Abelian, Geom. Dedicata 104 (2004), 71–96. MR 2043955, DOI 10.1023/B:GEOM.0000022864.30278.34
- M. Bestvina, M. Feighn, and M. Handel, The Tits alternative for $\textrm {Out}(F_ n)$. II. A Kolchin type theorem, Ann. of Math. 161 (2005), no. 1, 1–59.
- 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
- Hyman Bass and Alexander Lubotzky, Linear-central filtrations on groups, The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992) Contemp. Math., vol. 169, Amer. Math. Soc., Providence, RI, 1994, pp. 45–98. MR 1292897, DOI 10.1090/conm/169/01651
- Brian H. Bowditch, Tight geodesics in the curve complex, Invent. Math. 171 (2008), no. 2, 281–300. MR 2367021, DOI 10.1007/s00222-007-0081-y
- 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
- Albert Fathi, François Laudenbach, and Valentin Poénaru, Thurston’s work on surfaces, Mathematical Notes, vol. 48, Princeton University Press, Princeton, NJ, 2012. Translated from the 1979 French original by Djun M. Kim and Dan Margalit. MR 3053012
- Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR 2850125
- M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
- R. Gupta and D. Wigglesworth, Loxodromic elements in the cyclic splitting complex and their centralizers, arXiv:1710.10478 (2017).
- M. Handel and L. Mosher, Lipschitz retraction and distortion for subgroups of $\mathsf {Out}(F_ n)$, arXiv:1009.5018v1 (2010).
- Michael Handel and Lee Mosher, Lipschitz retraction and distortion for subgroups of $\textrm {Out}(F_n)$, Geom. Topol. 17 (2013), no. 3, 1535–1579. MR 3073930, DOI 10.2140/gt.2013.17.1535
- M. Handel and L. Mosher, The free splitting complex of a free group, I: Hyperbolicity, Geom. Topol. 17 (2013), 1581–1670.
- M. Handel and L. Mosher, Hyperbolic actions and 2nd bounded cohomology of subgroups of $\mathsf {Out}(F_n)$. Part I: Infinite lamination subgroups, arXiv:1511.06913 (2015).
- M. Handel and L. Mosher, Hyperbolic actions and 2nd bounded cohomology of subgroups of $\mathsf {Out}(F_n)$. Part II: Finite lamination subgroups (in preparation).
- M. Handel and L. Mosher, Virtually abelian subgroups of $\text {IA}_n(\mathbb {Z}/3)$ are abelian, arXiv:1801.09241 (2018).
- M. Handel and L. Mosher, Subgroup decomposition in $\mathsf {Out}(F_n)$, Memoirs AMS (to appear). See also Intro, arXiv:1302.2681 (2013); Part I, arXiv:1302.2378 (2013); Part II, arXiv:1302.2379 (2013); Part III, arXiv:1306.4712 (2013); Part IV, arXiv:1306.4711 (2013).
- Ilya Kapovich and Martin Lustig, Stabilizers of $\Bbb R$-trees with free isometric actions of $F_N$, J. Group Theory 14 (2011), no. 5, 673–694. MR 2831965, DOI 10.1515/JGT.2010.070
- 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
- Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504, DOI 10.1007/978-3-642-61856-7
- Peter Scott and Terry Wall, Topological methods in group theory, Homological group theory (Proc. Sympos., Durham, 1977) London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 137–203. MR 564422
Additional Information
- Michael Handel
- Affiliation: Department of Mathematics, The City University of New York, New York, New York 10016
- MR Author ID: 223960
- Email: michael.handel@lehman.cuny.edu
- Lee Mosher
- Affiliation: Department of Mathematics and Computer Science, Rutgers University, Newark, New Jersey 07102
- MR Author ID: 248017
- Email: mosher@rutgers.edu
- Received by editor(s): June 9, 2017
- Received by editor(s) in revised form: July 23, 2018
- Published electronically: February 11, 2019
- Additional Notes: The first author was supported by NSF grant DMS-1308710 and various PSC-CUNY grants.
The second author was supported by NSF grant DMS-1406376. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 4053-4105
- MSC (2010): Primary 20F65, 57M07; Secondary 20F28, 20E05
- DOI: https://doi.org/10.1090/tran/7698
- MathSciNet review: 4009387