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Connectivity of the space of ending laminations

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Leininger, Christopher J. and Schleimer, Saul. (2009) Connectivity of the space of ending laminations. Duke Mathematical Journal, Vol.150 (No.3). pp. 533-575. ISSN 0012-7094

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Official URL: http://dx.doi.org/10.1215/00127094-2009-059

Abstract

We prove that for any closed surface of genus at least four, and any punctured surface of genus at least two, the space of ending laminations is connected. A theorem of E. Klarreich [28, Theorem 1.3] implies that this space is homeomorphic to the Gromov boundary of the complex of curves. It follows that the boundary of the complex of curves is connected in these cases, answering the conjecture of P. Storm. Other applications include the rigidity of the complex of curves and connectivity of spaces of degenerate Kleinian groups.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Surfaces, Algebraic, Hyperbolic spaces, Curves, Algebraic
Journal or Publication Title:	Duke Mathematical Journal
Publisher:	Duke University Press
ISSN:	0012-7094
Date:	2009
Volume:	Vol.150
Number:	No.3
Page Range:	pp. 533-575
Identification Number:	10.1215/00127094-2009-059
Status:	Peer Reviewed
Access rights to Published version:	Open Access
Funder:	National Science Foundation (U.S.) (NSF)
Grant number:	DMS-0603881 (NSF), DMS-0508971 (NSF)
References:	[1] J. W. ANDERSON and R. D. CANARY, Cores of hyperbolic 3-manifolds and limits of Kleinian groups, II, J. London Math. Soc. (2) 61 (2000), 489 – 505. MR 1760675 [2] R. BENEDETTI and C. PETRONIO, Lectures on Hyperbolic Geometry, Universitext, Springer, Berlin, 1992. MR 1219310 [3] L. BERS, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94 – 97. MR 0111834 [4] ———, Fiber spaces over Teichm¨uller spaces, Acta. Math. 130 (1973), 89 – 126. MR 0430318 [5] M. BESTVINA, Questions in geometric group theory, preprint, 2004. [6] J. S. BIRMAN, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969), 213 – 238. MR 0243519 [7] ———, Braids, Links, and Mapping Class Groups, Ann. of Math. Stud. 82, Princeton Univ. Press, Princeton, 1974. MR 0375281 [8] F. BONAHON, Bouts des vari´et´es hyperboliques de dimension 3, Ann. of Math. (2) 124 (1986), 71 – 158. MR 0847953 [9] B. H. BOWDITCH, Intersection numbers and the hyperbolicity of the curve complex, J. Reine Angew. Math. 598 (2006), 105 – 129. MR 2270568 [10] J. F. BROCK, Continuity of Thurston’s length function, Geom. Funct. Anal. 10 (2000), 741 – 797. MR 1791139 [11] J. F. BROCK and K. W. BROMBERG, On the density of geometrically finite Kleinian groups, Acta Math. 192 (2004), 33 – 93. MR 2079598 [12] J. F. BROCK, R. D. CANARY, and Y. N. MINSKY, The classification of Kleinian surface groups, II: The ending lamination conjecture, preprint, arXiv:math/0412006v1 [math.GT] [13] R. D. CANARY, A covering theorem for hyperbolic 3-manifolds and its applications, Topology 35 (1996), 751 – 778. MR 1396777 [14] R. D. CANARY, D. B. A. EPSTEIN, and P. GREEN, “Notes on notes of Thurston” in Analytical and Geometric Aspects of Hyperbolic Space (Coventry/Durham, England, 1984), London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press, Cambridge, 1987, 3 – 92. MR 0903850 [15] C. J. EARLE and J. EELLS, A fibre bundle description of Teichm¨uller theory, J. Differential Geometry 3 (1969), 19 – 43. MR 0276999 [16] A. ESKIN, H. MASUR, and A. ZORICH, Moduli spaces of abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. Inst. Hautes ´ Etudes Sci. 97 (2003), 61 – 179. MR 2010740 [17] B. FARB and L. MOSHER, Convex cocompact subgroups of mapping class groups, Geom. Topol. 6 (2002), 91 – 152. MR 1914566 [18] A. FATHI, F. LAUDENBACH, and V. PO´ENARU, eds., Travaux de Thurston sur les surfaces, reprint of the 1979 ed., Ast´erisque 66 – 67, S´eminaire Orsay, Soc. Math. France, Montrouge, 1991. MR 1134426 [19] D. GABAI, Almost filling laminations and the connectivity of ending lamination space, Geom. Topol. 13 (2009), 1017 – 1041. MR 2470969 [20] F. P. GARDINER, Teichm¨uller Theory and Quadratic Differentials, Pure Appl. Math. (N.Y.), Wiley-Interscience, New York, 1987. MR 0903027 [21] U. HAMENST¨ADT, “Train tracks and the Gromov boundary of the complex of curves” in Spaces of Kleinian Groups, London Math. Soc. Lecture Note Ser. 329, Cambridge Univ. Press, Cambridge, 2006, 187 – 207. MR 2258749 [22] J. HUBBARD and H. MASUR, Quadratic differentials and foliations, Acta Math. 142 (1979), 221 – 274. MR 0523212 [23] S. P. HUMPHRIES, “Generators for the mapping class group” in Topology of Low-Dimensional Manifolds (Chelwood Gate, England, 1977), Lecture Notes in Math. 722, Springer, Berlin, 1979, 44 – 47. MR 0547453 [24] N. V. IVANOV, Subgroups of Teichm¨uller Modular Groups, revised by the author, trans. by E. J. F. Primrose, Trans. Math. Monogr. 115, Amer. Math. Soc., Providence, 1992. MR 1195787 [25] R. P. KENT IV and C. J. LEININGER, Shadows of mapping class groups: Capturing convex cocompactness, Geom. Funct. Anal. 18 (2008), 1270 – 1325. MR 2465691 [26] R. P. KENT IV, C. J. LEININGER, and S. SCHLEIMER, Trees and mapping class groups, preprint, arXiv:math/0611241v2 [math.GT] [27] S. KERCKHOFF, H. MASUR, and J. SMILLIE, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2) 124 (1986), 293 – 311. MR 0855297 [28] E. KLARREICH, The boundary at infinity of the curve complex and the relative Teichm¨uller space, preprint, 1999. [29] R. S. KULKARNI and P. B. SHALEN, On Ahlfors’ finiteness theorem, Adv. Math. 76 (1989), 155 – 169. MR 1013665 [30] C. J. LEININGER, M. MJ, and S. SCHLEIMER, The universal Cannon-Thurston map and the boundary of the curve complex, preprint, arXiv:0808.3521v2 [math.GT] [31] A. MARDEN and K. STREBEL, The heights theorem for quadratic differentials on Riemann surfaces, Acta Math. 153 (1984), 153 – 211. MR 0766263 [32] H. A. MASUR and Y. N. MINSKY, Geometry of the complex of curves, I: Hyperbolicity, Invent. Math. 138 (1999), 103 – 149. MR 1714338 [33] D. MCCULLOUGH, Compact submanifolds of 3-manifolds with boundary, Quart. J. Math. Oxford Ser. (2) 37 (1986), 299 – 307. MR 0854628 [34] Y. N. MINSKY, “Curve complexes, surfaces and 3-manifolds” in International Congress of Mathematicians, Vol. II, Eur. Math. Soc., Z¨urich, 2006, 1001 – 1033. MR 2275633 [35] ———, The classification of Kleinian surface groups, I: Models and bounds, preprint, arXiv:math/0302208v3 [math.GT] [36] M. MITRA, Cannon-Thurston maps for hyperbolic group extensions, Topology 37 (1998), 527 – 538. MR 1604882 [37] L. MOSHER, Train track expansions of measured foliations, preprint, 2003. [38] J.-P. OTAL, The Hyperbolization Theorem for Fibered 3-Manifolds, translated from the 1996 French original by Leslie D. Kay, SMF/AMS Texts Monogr. 7, Amer. Math. Soc., Providence, 2001. MR 1855976 [39] K. RAFI and S. SCHLEIMER, Curve complexes with connected boundary are rigid, preprint, arXiv:0710.3794v1 [math.GT] [40] G. P. SCOTT, Compact submanifolds of 3-manifolds, J. London Math. Soc. (2) 7 (1973), 246 – 250. MR 0326737 [41] W. P. THURSTON, Three-Dimensional Geometry and Topology, Vol. 1, Princeton Math. Ser. 35, Princeton Univ. Press, Princeton, 1997. MR 1435975 [42] ———, The geometry and topology of 3-manifolds, Princeton lecture notes, 1979. [43] ———, Hyperbolic structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle, preprint, arXiv:math/9801045v1 [math.GT]
URI:	http://wrap.warwick.ac.uk/id/eprint/3138