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Hyperbolic Actions and 2nd Bounded Cohomology of Subgroups of ${\mathsf {Out}}(F_n)$
About this Title
Michael Handel and Lee Mosher
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 292, Number 1454
ISBNs: 978-1-4704-6698-5 (print); 978-1-4704-7699-1 (online)
DOI: https://doi.org/10.1090/memo/1454
Published electronically: November 21, 2023
Table of Contents
1. Infinite Lamination Subgroups
- 1. Introduction
- 2. Background material
- 3. Reducing Theorem A to Theorem C and the WWPD Construction Theorem
- 4. Well functions and weak tiling functions
- 5. Proof of the WWPD Construction Theorem
2. Finite Lamination Subgroups
- 6. Introduction
- 7. Lifting to an automorphism group
- 8. Hyperbolic Action Theorem, Multi-edge case: Introduction
- 9. Flaring in a top EG stratum
- 10. Flaring in $T^*$ and hyperbolicity of $\mathcal {S}$
- 11. Abelian subgroups of $\mathsf {Out}(F_n)$
- 12. A train track semigroup action
- 13. The suspension action
Abstract
In this two part work we prove that for every finitely generated subgroup $\Gamma <{\mathsf {Out}}(F_n)$, either $\Gamma$ is virtually abelian or $H^2_b(\Gamma ;{\mathbb {R}})$ contains a vector space embedding of $\ell ^1$. The method uses actions on hyperbolic spaces. In Part I we focus on the case of infinite lamination subgroups $\Gamma$—those for which the set of all attracting laminations of all elements of $\Gamma$ is an infinite set—using actions on free splitting complexes of free groups. In Part II we focus on finite lamination subgroups $\Gamma$ and on the construction of useful new hyperbolic actions of those subgroups.- 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
- 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, Ken Bromberg, and Koji Fujiwara, Stable commutator length on mapping class groups, Ann. Inst. Fourier (Grenoble) 66 (2016), no. 3, 871–898 (English, with English and French summaries). MR 3494163
- Jason A. Behrstock, Asymptotic geometry of the mapping class group and Teichmüller space, Geom. Topol. 10 (2006), 1523–1578. MR 2255505, DOI 10.2140/gt.2006.10.1523
- M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992), no. 1, 85–101. MR 1152226
- Martin R. Bridson and Benson Farb, A remark about actions of lattices on free groups, Topology Appl. 110 (2001), no. 1, 21–24. Geometric topology and geometric group theory (Milwaukee, WI, 1997). MR 1804693, DOI 10.1016/S0166-8641(99)00174-1
- 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, A hyperbolic $\textrm {Out}(F_n)$-complex, Groups Geom. Dyn. 4 (2010), no. 1, 31–58. MR 2566300, DOI 10.4171/GGD/74
- 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 ${\mathrm {Out}}(F_ n)$. 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
- Mladen Bestvina, Mark Feighn, and Michael Handel, The Tits alternative for ${\mathrm {Out}}(F_ n)$. 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
- Jason Behrstock, Mark F. Hagen, and Alessandro Sisto, Hierarchically hyperbolic spaces, I: Curve complexes for cubical groups, Geom. Topol. 21 (2017), no. 3, 1731–1804. MR 3650081, DOI 10.2140/gt.2017.21.1731
- Jason Behrstock, Bruce Kleiner, Yair Minsky, and Lee Mosher, Geometry and rigidity of mapping class groups, Geom. Topol. 16 (2012), no. 2, 781–888. MR 2928983, DOI 10.2140/gt.2012.16.781
- M. Burger and N. Monod, Bounded cohomology of lattices in higher rank Lie groups, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 2, 199–235. MR 1694584, DOI 10.1007/s100970050007
- Jason A. Behrstock and Yair N. Minsky, Dimension and rank for mapping class groups, Ann. of Math. (2) 167 (2008), no. 3, 1055–1077. MR 2415393, DOI 10.4007/annals.2008.167.1055
- Jason A. Behrstock and Yair N. Minsky, Centroids and the rapid decay property in mapping class groups, J. Lond. Math. Soc. (2) 84 (2011), no. 3, 765–784.
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
- 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
- Robert Brooks, Some remarks on bounded cohomology, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., No. 97, Princeton Univ. Press, Princeton, NJ, 1981, pp. 53–63. MR 624804
- Robert Brooks and Caroline Series, Bounded cohomology for surface groups, Topology 23 (1984), no. 1, 29–36. MR 721449, DOI 10.1016/0040-9383(84)90022-3
- Gilbert Baumslag and Tekla Taylor, The centre of groups with one defining relator, Math. Ann. 175 (1968), 315–319. MR 222144, DOI 10.1007/BF02063216
- Martin R. Bridson and Richard D. Wade, Actions of higher-rank lattices on free groups, Compos. Math. 147 (2011), no. 5, 1573–1580. MR 2834733, DOI 10.1112/S0010437X11005598
- Marc Culler and John W. Morgan, Group actions on $\textbf {R}$-trees, Proc. London Math. Soc. (3) 55 (1987), no. 3, 571–604. MR 907233, DOI 10.1112/plms/s3-55.3.571
- Marc Culler and Karen Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), no. 1, 91–119. MR 830040, DOI 10.1007/BF01388734
- Moon Duchin and Kasra Rafi, Divergence of geodesics in Teichmüller space and the mapping class group, Geom. Funct. Anal. 19 (2009), no. 3, 722–742. MR 2563768, DOI 10.1007/s00039-009-0017-3
- Spencer Dowdall and Samuel J. Taylor, The co-surface graph and the geometry of hyperbolic free group extensions, J. Topol. 10 (2017), no. 2, 447–482. MR 3653318, DOI 10.1112/topo.12013
- David B. A. Epstein and Koji Fujiwara, The second bounded cohomology of word-hyperbolic groups, Topology 36 (1997), no. 6, 1275–1289. MR 1452851, DOI 10.1016/S0040-9383(96)00046-8
- Mark Feighn and Michael Handel, Abelian subgroups of $\textrm {Out}(F_n)$, Geom. Topol. 13 (2009), no. 3, 1657–1727. MR 2496054, DOI 10.2140/gt.2009.13.1657
- 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 Lee Mosher, Convex cocompact subgroups of mapping class groups, Geom. Topol. 6 (2002), 91–152. MR 1914566, DOI 10.2140/gt.2002.6.91
- Max Forester, Deformation and rigidity of simplicial group actions on trees, Geom. Topol. 6 (2002), 219–267. MR 1914569, DOI 10.2140/gt.2002.6.219
- Koji Fujiwara, The second bounded cohomology of a group acting on a Gromov-hyperbolic space, Proc. London Math. Soc. (3) 76 (1998), no. 1, 70–94. MR 1476898, DOI 10.1112/S0024611598000033
- Koji Fujiwara, The second bounded cohomology of an amalgamated free product of groups, Trans. Amer. Math. Soc. 352 (2000), no. 3, 1113–1129. MR 1491864, DOI 10.1090/S0002-9947-99-02282-5
- Vincent Guirardel and Camille Horbez, Boundaries of relative factor graphs and subgroup classification for automorphisms of free products, Geom. Topol. 26 (2022), no. 1, 71–126. MR 4404875, DOI 10.2140/gt.2022.26.71
- P. Ghosh, Applications of weak attraction theory in $\mathsf {Out}(F_n)$, arXiv:1306.6049, 2013.
- Pritam Ghosh, Applications of weak attraction theory in $\textrm {Out}(\Bbb {F}_N)$, Geom. Dedicata 181 (2016), 1–22. MR 3475735, DOI 10.1007/s10711-015-0109-1
- Damien Gaboriau, Andre Jaeger, Gilbert Levitt, and Martin Lustig, An index for counting fixed points of automorphisms of free groups, Duke Math. J. 93 (1998), no. 3, 425–452. MR 1626723, DOI 10.1215/S0012-7094-98-09314-0
- Vincent Guirardel and Gilbert Levitt, Deformation spaces of trees, Groups Geom. Dyn. 1 (2007), no. 2, 135–181. MR 2319455, DOI 10.4171/GGD/8
- 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}
- Radhika Gupta, Loxodromic elements for the relative free factor complex, Geom. Dedicata 196 (2018), 91–121. MR 3853630, DOI 10.1007/s10711-017-0310-5
- Ursula Hamenstädt, Lines of minima in outer space, Duke Math. J. 163 (2014), no. 4, 733–776. MR 3178431, DOI 10.1215/00127094-2429807
- Matthias Hamann, Group actions on metric spaces: fixed points and free subgroups, Abh. Math. Semin. Univ. Hambg. 87 (2017), no. 2, 245–263. MR 3696149, DOI 10.1007/s12188-016-0164-z
- Michael Handel and Lee Mosher, Second bounded cohomology and WWPD, Kyoto J. Math. 61 (2021), no. 4, 873–904. MR 4415399, DOI 10.1215/21562261-2021-0017
- M. Handel and L. Mosher, Subgroup classification in $\text {Out}(F_n)$, arXiv:0908.1255, 2009.
- 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
- Michael Handel and Lee Mosher, The free splitting complex of a free group I: Hyperbolicity, Geom. Topol. 17 (2013), 1581–1670.
- M. Handel and L. Mosher, Hyperbolicity of relative free splitting and free factor complexes, arXiv:1407.3508, 2014.
- Michael Handel and Lee Mosher, The free splitting complex of a free group, II: Loxodromic outer automorphisms, Trans. Amer. Math. Soc. 372 (2019), no. 6, 4053–4105. MR 4009387, DOI 10.1090/tran/7698
- Michael Handel and Lee Mosher, Subgroup decomposition in $\mathsf {Out}(F_n)$, Memoirs AMS 898 (2020), no. 1280.
- Michael Handel and Lee Mosher, Virtually abelian subgroups of $\textrm {IA}_n(\Bbb {Z}/3)$ are abelian, Michigan Math. J. 69 (2020), no. 3, 465–485. MR 4132599, DOI 10.1307/mmj/1596700814
- Camille Horbez, A short proof of Handel and Mosher’s alternative for subgroups of Out$(F_N)$, Groups Geom. Dyn. 10 (2016), no. 2, 709–721. MR 3513113, DOI 10.4171/GGD/361
- Nikolai V. Ivanov, Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, vol. 115, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by E. J. F. Primrose and revised by the author. MR 1195787, DOI 10.1090/mmono/115
- Ilya Kapovich and Kasra Rafi, On hyperbolicity of free splitting and free factor complexes, Groups Geom. Dyn. 8 (2014), no. 2, 391–414. MR 3231221, DOI 10.4171/GGD/231
- Johanna Mangahas, Uniform uniform exponential growth of subgroups of the mapping class group, Geom. Funct. Anal. 19 (2010), no. 5, 1468–1480. MR 2585580, DOI 10.1007/s00039-009-0038-y
- G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825, DOI 10.1007/978-3-642-51445-6
- Mahan Mitra, Cannon-Thurston maps for trees of hyperbolic metric spaces, J. Differential Geom. 48 (1998), no. 1, 135–164. MR 1622603
- 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
- H. A. Masur and Y. N. Minsky, Geometry of the complex of curves. II. Hierarchical structure, Geom. Funct. Anal. 10 (2000), no. 4, 902–974. MR 1791145, DOI 10.1007/PL00001643
- Ashot Minasyan and Denis Osin, Acylindrical hyperbolicity of groups acting on trees, Math. Ann. 362 (2015), no. 3-4, 1055–1105. MR 3368093, DOI 10.1007/s00208-014-1138-z
- D. Witte Morris, Introduction to arithmetic groups, Deductive Press \verb+http://deductivepress.ca+, 2015.
- G. D. Mostow, Self-adjoint groups, Ann. of Math. (2) 62 (1955), 44–55. MR 69830, DOI 10.2307/2007099
- Mahan Mj and Pranab Sardar, A combination theorem for metric bundles, Geom. Funct. Anal. 22 (2012), no. 6, 1636–1707. MR 3000500, DOI 10.1007/s00039-012-0196-1
- D. Osin, Acylindrically hyperbolic groups, Trans. Amer. Math. Soc. 368 (2016), no. 2, 851–888. MR 3430352, DOI 10.1090/tran/6343
- Jing Tao, Linearly bounded conjugator property for mapping class groups, Geom. Funct. Anal. 23 (2013), no. 1, 415–466. MR 3037904, DOI 10.1007/s00039-012-0206-3
- Karen Vogtmann, Automorphisms of free groups and outer space, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), 2002, pp. 1–31. MR 1950871, DOI 10.1023/A:1020973910646
- 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
- Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR 776417, DOI 10.1007/978-1-4684-9488-4