The geometry of the disk complex
HTML articles powered by AMS MathViewer
- by Howard Masur and Saul Schleimer;
- J. Amer. Math. Soc. 26 (2013), 1-62
- DOI: https://doi.org/10.1090/S0894-0347-2012-00742-5
- Published electronically: August 22, 2012
Abstract:
We give a distance estimate for the disk complex. We use the distance estimate to prove that the disk complex is Gromov hyperbolic. As another application of our techniques, we find an algorithm which computes the Hempel distance of a Heegaard splitting, up to an error depending only on the genus.References
- Jason Behrstock. Asymptotic geometry of the mapping class group and Teichmüller space. Ph.D. thesis, SUNY Stony Brook, 2004. http://www.math.columbia.edu/$\sim$jason/thesis.pdf.
- Jason Behrstock, Cornelia Druţu, and Lee Mosher, Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity, Math. Ann. 344 (2009), no. 3, 543–595. MR 2501302, DOI 10.1007/s00208-008-0317-1
- Mladen Bestvina and Koji Fujiwara, Quasi-homomorphisms on mapping class groups, Glas. Mat. Ser. III 42(62) (2007), no. 1, 213–236. MR 2332668, DOI 10.3336/gm.42.1.15
- Joan S. Birman, The topology of 3-manifolds, Heegaard distance and the mapping class group of a 2-manifold, Problems on mapping class groups and related topics, Proc. Sympos. Pure Math., vol. 74, Amer. Math. Soc., Providence, RI, 2006, pp. 133–149. MR 2264538, DOI 10.1090/pspum/074/2264538
- Francis Bonahon, Cobordism of automorphisms of surfaces, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 237–270. MR 732345, DOI 10.24033/asens.1448
- B. H. Bowditch, A short proof that a subquadratic isoperimetric inequality implies a linear one, Michigan Math. J. 42 (1995), no. 1, 103–107. MR 1322192, DOI 10.1307/mmj/1029005156
- 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
- Tara E. Brendle and Dan Margalit, Commensurations of the Johnson kernel, Geom. Topol. 8 (2004), 1361–1384. MR 2119299, DOI 10.2140/gt.2004.8.1361
- 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
- Jeffrey F. Brock, The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores, J. Amer. Math. Soc. 16 (2003), no. 3, 495–535. MR 1969203, DOI 10.1090/S0894-0347-03-00424-7
- Alberto Cavicchioli and Fulvia Spaggiari, A note on irreducible Heegaard diagrams, Int. J. Math. Math. Sci. , posted on (2006), Art. ID 53135, 11. MR 2251669, DOI 10.1155/IJMMS/2006/53135
- Young-Eun Choi and Kasra Rafi, Comparison between Teichmüller and Lipschitz metrics, J. Lond. Math. Soc. (2) 76 (2007), no. 3, 739–756. MR 2377122, DOI 10.1112/jlms/jdm052
- H. Cišang, Simple path systems on full pretzels, Mat. Sb. (N.S.) 66(108) (1965), 230–239 (Russian). MR 193633
- M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990 (French). Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups]; With an English summary. MR 1075994, DOI 10.1007/BFb0084913
- Robert H. Gilman, The geometry of cycles in the Cayley diagram of a group, 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. 331–340. MR 1292908, DOI 10.1090/conm/169/01663
- Robert H. Gilman, On the definition of word hyperbolic groups, Math. Z. 242 (2002), no. 3, 529–541. MR 1985464, DOI 10.1007/s002090100356
- C. McA. Gordon, $3$-dimensional topology up to 1960, History of topology, North-Holland, Amsterdam, 1999, pp. 449–489. MR 1674921, DOI 10.1016/B978-044482375-5/50016-X
- 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}
- Wolfgang Haken, Various aspects of the three-dimensional Poincaré problem, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) Markham Publishing Co., Chicago, IL, 1970, pp. 140–152. MR 273624
- Ursula Hamenstädt, Geometry of the mapping class groups. I. Boundary amenability, Invent. Math. 175 (2009), no. 3, 545–609. MR 2471596, DOI 10.1007/s00222-008-0158-2
- Kevin Hartshorn, Heegaard splittings of Haken manifolds have bounded distance, Pacific J. Math. 204 (2002), no. 1, 61–75. MR 1905192, DOI 10.2140/pjm.2002.204.61
- W. J. Harvey, Boundary structure of the modular group, 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. 245–251. MR 624817
- A. Hatcher and W. Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980), no. 3, 221–237. MR 579573, DOI 10.1016/0040-9383(80)90009-9
- John Hempel, $3$-Manifolds, Annals of Mathematics Studies, No. 86, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1976. MR 415619
- John Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (2001), no. 3, 631–657. MR 1838999, DOI 10.1016/S0040-9383(00)00033-1
- John Hamal Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, 2006. Teichmüller theory; With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra; With forewords by William Thurston and Clifford Earle. MR 2245223
- Tsuyoshi Kobayashi, Heights of simple loops and pseudo-Anosov homeomorphisms, Braids (Santa Cruz, CA, 1986) Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 327–338. MR 975087, DOI 10.1090/conm/078/975087
- Jason Leasure. Geodesics in the complex of curves of a surface. Ph.D. thesis. http://repositories.lib.utexas.edu/bitstream/handle/2152/1700/leasurejp46295.pdf.
- 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
- 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
- Howard A. Masur and Yair N. Minsky, Quasiconvexity in the curve complex, In the tradition of Ahlfors and Bers, III, Contemp. Math., vol. 355, Amer. Math. Soc., Providence, RI, 2004, pp. 309–320. MR 2145071, DOI 10.1090/conm/355/06460
- Howard A. Masur, Lee Mosher, and Saul Schleimer. On train track splitting sequences. http://arXiv:1004.4564v1.
- Darryl McCullough, Virtually geometrically finite mapping class groups of $3$-manifolds, J. Differential Geom. 33 (1991), no. 1, 1–65. MR 1085134
- 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
- Lee Mosher. Train track expansions of measured foliations. 2003. http://newark.rutgers. edu/$\sim$mosher/.
- Subhashis Nag, The complex analytic theory of Teichmüller spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. MR 927291
- R. C. Penner and J. L. Harer, Combinatorics of train tracks, Annals of Mathematics Studies, vol. 125, Princeton University Press, Princeton, NJ, 1992. MR 1144770, DOI 10.1515/9781400882458
- Robert C. Penner, A construction of pseudo-Anosov homeomorphisms, Trans. Amer. Math. Soc. 310 (1988), no. 1, 179–197. MR 930079, DOI 10.1090/S0002-9947-1988-0930079-9
- Kasra Rafi, A combinatorial model for the Teichmüller metric, Geom. Funct. Anal. 17 (2007), no. 3, 936–959. MR 2346280, DOI 10.1007/s00039-007-0615-x
- Kasra Rafi. Hyperbolicity in Teichmüller space. November 2010, http://arXiv:1011.6004.
- Kasra Rafi and Saul Schleimer, Covers and the curve complex, Geom. Topol. 13 (2009), no. 4, 2141–2162. MR 2507116, DOI 10.2140/gt.2009.13.2141
- Dale Rolfsen, Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original. MR 1277811
- Martin Scharlemann, The complex of curves on nonorientable surfaces, J. London Math. Soc. (2) 25 (1982), no. 1, 171–184. MR 645874, DOI 10.1112/jlms/s2-25.1.171
- Saul Schleimer. Notes on the complex of curves. http://www.warwick.ac.uk/ masgar/Maths/notes.pdf.
- Kenneth J. Shackleton. Tightness and computing distances in the curve complex, http://arxiv.org/abs/math/0412078v3.
- 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
- Friedhelm Waldhausen, Some problems on $3$-manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, RI, 1978, pp. 313–322. MR 520549
- Heiner Zieschang, On Heegaard diagrams of $3$-manifolds, Astérisque 163-164 (1988), 7, 247–280, 283 (1989) (English, with French summary). On the geometry of differentiable manifolds (Rome, 1986). MR 999976
Bibliographic Information
- Howard Masur
- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
- MR Author ID: 219274
- Email: masur@math.uic.edu
- Saul Schleimer
- Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
- MR Author ID: 689853
- Email: s.schleimer@warwick.ac.uk
- Received by editor(s): November 23, 2010
- Received by editor(s) in revised form: April 6, 2012
- Published electronically: August 22, 2012
- Additional Notes: This work is in the public domain.
- Journal: J. Amer. Math. Soc. 26 (2013), 1-62
- MSC (2010): Primary 57M50
- DOI: https://doi.org/10.1090/S0894-0347-2012-00742-5
- MathSciNet review: 2983005