A universal Cannon-Thurston map and the surviving curve complex

Gültepe, Funda; Leininger, Christopher; Pho-on, Witsarut

doi:10.1090/btran/99

A universal Cannon-Thurston map and the surviving curve complex
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by Funda Gültepe, Christopher J. Leininger and Witsarut Pho-on HTML | PDF

Trans. Amer. Math. Soc. Ser. B 9 (2022), 99-143

Abstract:

Using the Birman exact sequence for pure mapping class groups, we construct a universal Cannon-Thurston map onto the boundary of a curve complex for a surface with punctures we call surviving curve complex. Along the way we prove hyperbolicity of this complex and identify its boundary as a space of laminations. As a corollary we obtain a universal Cannon-Thurston map to the boundary of the ordinary curve complex, extending earlier work of the second author with Mj and Schleimer [Comment. Math. Helv. 86 (2011), pp. 769–816].

References

R. C. Alperin, Warren Dicks, and J. Porti, The boundary of the Gieseking tree in hyperbolic three-space, Topology Appl. 93 (1999), no. 3, 219–259. MR 1688476, DOI 10.1016/S0166-8641(97)00270-8
Tarik Aougab, Uniform hyperbolicity of the graphs of curves, Geom. Topol. 17 (2013), no. 5, 2855–2875. MR 3190300, DOI 10.2140/gt.2013.17.2855
Benjamin Beeker, Matthew Cordes, Giles Gardam, Radhika Gupta, and Emily Stark, Cannon–Thurston maps for CAT(0) groups with isolated flats, arXiv:1810.13285 to appear in Math. Ann., 2018.
Jeffrey F. Brock, Richard D. Canary, and Yair N. Minsky, The classification of Kleinian surface groups, II: The ending lamination conjecture, Ann. of Math. (2) 176 (2012), no. 1, 1–149. MR 2925381, DOI 10.4007/annals.2012.176.1.1
Jason Alan Behrstock, Asymptotic geometry of the mapping class group and Teichmuller space, ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Ph.D.)–State University of New York at Stony Brook. MR 2705954
M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992), no. 1, 85–101. MR 1152226, DOI 10.4310/jdg/1214447806
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
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
Francis Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. (2) 124 (1986), no. 1, 71–158 (French). MR 847953, DOI 10.2307/1971388
Brian H. Bowditch, Intersection numbers and the hyperbolicity of the curve complex, J. Reine Angew. Math. 598 (2006), 105–129. MR 2270568, DOI 10.1515/CRELLE.2006.070
Brian H. Bowditch, The Cannon-Thurston map for punctured-surface groups, Math. Z. 255 (2007), no. 1, 35–76. MR 2262721, DOI 10.1007/s00209-006-0012-4
B. H. Bowditch, Stacks of hyperbolic spaces and ends of 3-manifolds, Geometry and topology down under, Contemp. Math., vol. 597, Amer. Math. Soc., Providence, RI, 2013, pp. 65–138. MR 3186670, DOI 10.1090/conm/597/11769
Brian H. Bowditch, Uniform hyperbolicity of the curve graphs, Pacific J. Math. 269 (2014), no. 2, 269–280. MR 3238474, DOI 10.2140/pjm.2014.269.269
O. Baker and T. R. Riley, Cannon-Thurston maps do not always exist, Forum Math. Sigma 1 (2013), Paper No. e3, 11. MR 3143716, DOI 10.1017/fms.2013.4
Owen Baker and Timothy Riley, Cannon-Thurston maps, subgroup distortion, and hyperbolic hydra, Groups Geom. Dyn. 14 (2020), no. 1, 255–282. MR 4077662, DOI 10.4171/ggd/543
Andrew J. Casson and Steven A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts, vol. 9, Cambridge University Press, Cambridge, 1988. MR 964685, DOI 10.1017/CBO9780511623912
R. D. Canary, D. B. A. Epstein, and P. L. Green, Notes on notes of Thurston [MR0903850], Fundamentals of hyperbolic geometry: selected expositions, London Math. Soc. Lecture Note Ser., vol. 328, Cambridge Univ. Press, Cambridge, 2006, pp. 1–115. With a new foreword by Canary. MR 2235710
Matt T. Clay, Christopher J. Leininger, and Johanna Mangahas, The geometry of right-angled Artin subgroups of mapping class groups, Groups Geom. Dyn. 6 (2012), no. 2, 249–278. MR 2914860, DOI 10.4171/GGD/157
Matt Clay, Kasra Rafi, and Saul Schleimer, Uniform hyperbolicity of the curve graph via surgery sequences, Algebr. Geom. Topol. 14 (2014), no. 6, 3325–3344. MR 3302964, DOI 10.2140/agt.2014.14.3325
James W. Cannon and William P. Thurston, Group invariant Peano curves, Geom. Topol. 11 (2007), 1315–1355. MR 2326947, DOI 10.2140/gt.2007.11.1315
Matthew Gentry Durham, Mark F. Hagen, and Alessandro Sisto, Boundaries and automorphisms of hierarchically hyperbolic spaces, Geom. Topol. 21 (2017), no. 6, 3659–3758. MR 3693574, DOI 10.2140/gt.2017.21.3659
Spencer Dowdall, Ilya Kapovich, and Samuel J. Taylor, Cannon-Thurston maps for hyperbolic free group extensions, Israel J. Math. 216 (2016), no. 2, 753–797. MR 3557464, DOI 10.1007/s11856-016-1426-2
Shubhabrata Das and Mahan Mj, Semiconjugacies between relatively hyperbolic boundaries, Groups Geom. Dyn. 10 (2016), no. 2, 733–752. MR 3513115, DOI 10.4171/GGD/363
Sérgio R. Fenley, Asymptotic properties of depth one foliations in hyperbolic $3$-manifolds, J. Differential Geom. 36 (1992), no. 2, 269–313. MR 1180384
Sérgio R. Fenley, Quasigeodesic pseudo-Anosov flows in hyperbolic 3-manifolds and connections with large scale geometry, Adv. Math. 303 (2016), 192–278. MR 3552525, DOI 10.1016/j.aim.2016.05.015
Elizabeth Field, Trees, dendrites and the Cannon-Thurston map, Algebr. Geom. Topol. 20 (2020), no. 6, 3083–3126. MR 4185936, DOI 10.2140/agt.2020.20.3083
William J. Floyd, Group completions and limit sets of Kleinian groups, Invent. Math. 57 (1980), no. 3, 205–218. MR 568933, DOI 10.1007/BF01418926
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
Steven Frankel, Quasigeodesic flows and sphere-filling curves, Geom. Topol. 19 (2015), no. 3, 1249–1262. MR 3352235, DOI 10.2140/gt.2015.19.1249
François Guéritaud. Veering triangulations and Cannon-Thurston maps. J. Topol., 9(3):957–983, 2016.
Ursula Hamenstädt, Train tracks and the Gromov boundary of the complex of curves, Spaces of Kleinian groups, London Math. Soc. Lecture Note Ser., vol. 329, Cambridge Univ. Press, Cambridge, 2006, pp. 187–207. MR 2258749
Ursula Hamenstädt, Geometry of the complex of curves and of Teichmüller space, Handbook of Teichmüller theory. Vol. I, IRMA Lect. Math. Theor. Phys., vol. 11, Eur. Math. Soc., Zürich, 2007, pp. 447–467. MR 2349677, DOI 10.4171/029-1/11
John L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986), no. 1, 157–176. MR 830043, DOI 10.1007/BF01388737
Sebastian Hensel, Piotr Przytycki, and Richard C. H. Webb, 1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 4, 755–762. MR 3336835, DOI 10.4171/JEMS/517
Allen Hatcher and Karen Vogtmann, Tethers and homology stability for surfaces, Algebr. Geom. Topol. 17 (2017), no. 3, 1871–1916. MR 3677942, DOI 10.2140/agt.2017.17.1871
Woojin Jeon, Ilya Kapovich, Christopher Leininger, and Ken’ichi Ohshika, Conical limit points and the Cannon-Thurston map, Conform. Geom. Dyn. 20 (2016), 58–80. MR 3488025, DOI 10.1090/ecgd/294
Ilya Kapovich and Nadia Benakli, Boundaries of hyperbolic groups, Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001) Contemp. Math., vol. 296, Amer. Math. Soc., Providence, RI, 2002, pp. 39–93. MR 1921706, DOI 10.1090/conm/296/05068
Ilya Kapovich and Martin Lustig, Cannon-Thurston fibers for iwip automorphisms of $F_N$, J. Lond. Math. Soc. (2) 91 (2015), no. 1, 203–224. MR 3335244, DOI 10.1112/jlms/jdu069
Erica Klarreich, Semiconjugacies between Kleinian group actions on the Riemann sphere, Amer. J. Math. 121 (1999), no. 5, 1031–1078. MR 1713300, DOI 10.1353/ajm.1999.0034
Erica Klarreich, The boundary at infinity of the curve complex and the relative teichmueller space, http://nasw.org/users/klarreich/research.htm, Preprint, 1999.
Richard P. Kent IV, Christopher J. Leininger, and Saul Schleimer, Trees and mapping class groups, J. Reine Angew. Math. 637 (2009), 1–21. MR 2599078, DOI 10.1515/CRELLE.2009.087
Irwin Kra, On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces, Acta Math. 146 (1981), no. 3-4, 231–270. MR 611385, DOI 10.1007/BF02392465
Christopher J. Leininger, Accidental parabolics in the mapping class group, Proc. Amer. Math. Soc. 137 (2009), no. 3, 1153–1160. MR 2457458, DOI 10.1090/S0002-9939-08-09604-4
Christopher J. Leininger, Mahan Mj, and Saul Schleimer, The universal Cannon-Thurston map and the boundary of the curve complex, Comment. Math. Helv. 86 (2011), no. 4, 769–816. MR 2851869, DOI 10.4171/CMH/240
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
Curtis T. McMullen, Local connectivity, Kleinian groups and geodesics on the blowup of the torus, Invent. Math. 146 (2001), no. 1, 35–91. MR 1859018, DOI 10.1007/PL00005809
Yair N. Minsky, On rigidity, limit sets, and end invariants of hyperbolic $3$-manifolds, J. Amer. Math. Soc. 7 (1994), no. 3, 539–588. MR 1257060, DOI 10.1090/S0894-0347-1994-1257060-3
Yair Minsky, The classification of Kleinian surface groups. I. Models and bounds, Ann. of Math. (2) 171 (2010), no. 1, 1–107. MR 2630036, DOI 10.4007/annals.2010.171.1
M. Mitra, Ending laminations for hyperbolic group extensions, Geom. Funct. Anal. 7 (1997), no. 2, 379–402. MR 1445392, DOI 10.1007/PL00001624
Mahan Mitra, Cannon-Thurston maps for hyperbolic group extensions, Topology 37 (1998), no. 3, 527–538. MR 1604882, DOI 10.1016/S0040-9383(97)00036-0
Mahan Mitra, Cannon-Thurston maps for trees of hyperbolic metric spaces, J. Differential Geom. 48 (1998), no. 1, 135–164. MR 1622603
Mahan Mj, Cannon-Thurston maps for surface groups, Ann. of Math. (2) 179 (2014), no. 1, 1–80. MR 3126566, DOI 10.4007/annals.2014.179.1.1
Mahan Mj, Ending laminations and Cannon-Thurston maps, Geom. Funct. Anal. 24 (2014), no. 1, 297–321. With an appendix by Shubhabrata Das and Mj. MR 3177384, DOI 10.1007/s00039-014-0263-x
Mahan Mj, Cannon-Thurston maps for Kleinian groups, Forum Math. Pi 5 (2017), e1, 49. MR 3652816, DOI 10.1017/fmp.2017.2
Mahan Mj, Cannon-Thurston maps, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 885–917. MR 3966793
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
Yoshifumi Matsuda and Shin-ichi Oguni, On Cannon-Thurston maps for relatively hyperbolic groups, J. Group Theory 17 (2014), no. 1, 41–47. MR 3176651, DOI 10.1515/jgt-2013-0024
Sarah C. Mousley, Nonexistence of boundary maps for some hierarchically hyperbolic spaces, Algebr. Geom. Topol. 18 (2018), no. 1, 409–439. MR 3748248, DOI 10.2140/agt.2018.18.409
Mahan Mj and Abhijit Pal, Relative hyperbolicity, trees of spaces and Cannon-Thurston maps, Geom. Dedicata 151 (2011), 59–78. MR 2780738, DOI 10.1007/s10711-010-9519-2
Mahan Mj and Kasra Rafi, Algebraic ending laminations and quasiconvexity, Algebr. Geom. Topol. 18 (2018), no. 4, 1883–1916. MR 3797060, DOI 10.2140/agt.2018.18.1883
Howard Masur and Saul Schleimer, The geometry of the disk complex, J. Amer. Math. Soc. 26 (2013), no. 1, 1–62. MR 2983005, DOI 10.1090/S0894-0347-2012-00742-5
Abhijit Pal, Relatively hyperbolic extensions of groups and Cannon-Thurston maps, Proc. Indian Acad. Sci. Math. Sci. 120 (2010), no. 1, 57–68. MR 2654898, DOI 10.1007/s12044-010-0009-0
Witsarut Pho-On, Infinite unicorn paths and Gromov boundaries, Groups Geom. Dyn. 11 (2017), no. 1, 353–370. MR 3641844, DOI 10.4171/GGD/399
William P. Thurston, Geometry and topology of $3$-manifolds, lecture notes, Princeton University, 1978.
William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6
Kate M. Vokes, Hierarchical hyperbolicity of graphs of multicurves, Preprint, arXiv:1711.03080.
Richard C. H. Webb, Uniform bounds for bounded geodesic image theorems, J. Reine Angew. Math. 709 (2015), 219–228. MR 3430880, DOI 10.1515/crelle-2013-0109