×

zbMATH — the first resource for mathematics

The classification of Kleinian surface groups. II: The Ending lamination conjecture. (English) Zbl 1253.57009
Thurston’s Ending Lamination Conjecture states that a hyperbolic 3-manifold \(M = \mathbb H^3/G\) with finitely generated fundamental group is determined, up to isometry, by its topological type and its end invariants. This is related also to Marden’s Tameness Conjecture which has been recently proved by Agol and Calegari-Gabai and states that each end of \(M\) is topologically tame, i.e. homeomorphic to the product \(S \times [0, \infty)\) of a surface \(S\) with the positive reals (and hence \(M\) is homeomorphic to the interior of a compact 3-manifold). The end invariant associated to such a tame end is then a conformal structure on the surface \(S\) (a point in the Teichmüller space of \(S\)) if the end is also geometrically tame (geometrically finite), and a geodesic lamination on \(S\) if it is geometrically infinite (but still topologically tame).
In the present paper, the Ending Lamination Conjecture is proved for the basic case of a Kleinian surface group \(G\) (i.e., \(G\) is isomorphic to the fundamental group of a compact surface). By the proof of the tameness conjecture in this case (due to Thurston and Bonahon), the associated hyperbolic 3-manifold \(M = \mathbb H^3/G\) is homeomorphic to a product \(\text{int}(S) \times \mathbb R\), for a compact orientable surface \(S\), so there are exactly two end invariants in this case: each is a geodesic lamination on some (possibly empty) subsurface of \(S\) and a conformal structure on the complementary surface. This implies then also a proof of the Ending Lamination Conjecture for incompressible ends of a general hyperbolic 3-manifold \(M\) as above. The first part of the proof of the Ending Lamination Conjecture for Kleinian surface groups appeared in a previous paper of Y. Minsky [Ann. Math. (2) 171, No. 1, 1–107 (2010; Zbl 1193.30063)], and the proof of the Ending Lamination Conjecture for the general case of not necessarily incompressible ends will appear in the third paper of the series (in the meantime, Bowditch, Rees and Soma have given alternate proofs of the Ending Lamination Conjecture in which various aspects have been simplified).
As explained in the present paper, the solution of the Ending Lamination Conjecture is also a crucial ingredient in a proof of the last of the three great conjectures on Kleinian groups, the Bers-Sullivan-Thurston Density Conjecture, in particular the results of the present paper imply this conjecture for Kleinian surface groups and for the case of incompressible ends.
Concerning the proofs, “The main technical result that leads to the Ending Lamination Theorem is the Bilipschitz Model Theorem, which gives a bilipschitz homeomorphism from a model manifold \(M_\nu\) to the hyperbolic manifold \(M = \mathbb H^3/\rho(\pi_1(S))\). The model \(M_\nu\) was constructed in the paper by Minsky cited above, and its crucial property is that it depends only on the two end invariants and not on the representation \(\rho: \pi_1(S) \to \text{PSL}_2(\mathbb C)\) itself.”

MSC:
57M50 General geometric structures on low-dimensional manifolds
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] I. Agol, ”Volume change under drilling,” Geom. Topol., vol. 6, pp. 905-916, 2002. · Zbl 1031.57014 · doi:10.2140/gt.2002.6.905 · emis:journals/UW/gt/GTVol6/paper27.abs.html · eudml:123264 · arxiv:math/0101138
[2] I. Agol, Tameness of hyperbolic 3-manifolds, 2004. · Zbl 1178.57017
[3] L. Ahlfors and L. Bers, ”Riemann’s mapping theorem for variable metrics,” Ann. of Math., vol. 72, pp. 385-404, 1960. · Zbl 0104.29902 · doi:10.2307/1970141
[4] L. Ahlfors, ”Finitely generated Kleinian groups,” Amer. J. Math., vol. 86, pp. 413-429, 1964. · Zbl 0133.04201 · doi:10.2307/2373173
[5] J. W. Anderson and R. D. Canary, ”Algebraic limits of Kleinian groups which rearrange the pages of a book,” Invent. Math., vol. 126, iss. 2, pp. 205-214, 1996. · Zbl 0874.57012 · doi:10.1007/s002220050094
[6] J. W. Anderson and R. D. Canary, ”Cores of hyperbolic \(3\)-manifolds and limits of Kleinian groups,” Amer. J. Math., vol. 118, iss. 4, pp. 745-779, 1996. · Zbl 0863.30048 · doi:10.1353/ajm.1996.0031 · muse.jhu.edu · arxiv:math/9903076
[7] J. W. Anderson, R. D. Canary, M. Culler, and P. B. Shalen, ”Free Kleinian groups and volumes of hyperbolic \(3\)-manifolds,” J. Differential Geom., vol. 43, iss. 4, pp. 738-782, 1996. · Zbl 0860.57011 · projecteuclid.org
[8] J. W. Anderson, R. D. Canary, and D. McCullough, ”The topology of deformation spaces of Kleinian groups,” Ann. of Math., vol. 152, iss. 3, pp. 693-741, 2000. · Zbl 0976.57016 · doi:10.2307/2661352 · www.math.princeton.edu · eudml:121464 · arxiv:math/9806079
[9] J. W. Anderson, ”Intersections of topologically tame subgroups of Kleinian groups,” J. Anal. Math., vol. 65, pp. 77-94, 1995. · Zbl 0832.30027 · doi:10.1007/BF02788766
[10] J. Behrstock, B. Kleiner, Y. Minsky, and L. Mosher, Geometry and rigidity of mapping class groups, 2008. · Zbl 1281.20045 · doi:10.2140/gt.2012.16.781 · arxiv:0801.2006
[11] R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry, New York: Springer-Verlag, 1992. · Zbl 0768.51018 · doi:10.1007/978-3-642-58158-8
[12] L. Bers, ”Spaces of Kleinian groups,” in Several Complex Variables, I, New York: Springer-Verlag, 1970, pp. 9-34. · Zbl 0211.10602
[13] F. Bonahon, ”Bouts des variétés hyperboliques de dimension \(3\),” Ann. of Math., vol. 124, iss. 1, pp. 71-158, 1986. · Zbl 0671.57008 · doi:10.2307/1971388
[14] B. H. Bowditch, ”Hyperbolic 3-manifolds and the geometry of the curve complex,” in European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, pp. 103-115. · Zbl 1081.53032 · doi:10.4171/009-1/7
[15] B. H. Bowditch, ”Intersection numbers and the hyperbolicity of the curve complex,” J. Reine Angew. Math., vol. 598, pp. 105-129, 2006. · Zbl 1119.32006 · doi:10.1515/CRELLE.2006.070
[16] B. H. Bowditch, ”Systems of bands in hyperbolic 3-manifolds,” Pacific J. Math., vol. 232, iss. 1, pp. 1-42, 2007. · Zbl 1154.57015 · doi:10.2140/pjm.2007.232.1 · pjm.math.berkeley.edu
[17] J. F. Brock and K. W. Bromberg, ”On the density of geometrically finite Kleinian groups,” Acta Math., vol. 192, iss. 1, pp. 33-93, 2004. · Zbl 1055.57020 · doi:10.1007/BF02441085 · arxiv:math/0212189
[18] J. Brock, R. Canary, and Y. Minsky, The classification of finitely-generated Kleinian groups.
[19] K. Bromberg, ”Projective structures with degenerate holonomy and the Bers density conjecture,” Ann. of Math., vol. 166, iss. 1, pp. 77-93, 2007. · Zbl 1137.30014 · doi:10.4007/annals.2007.166.77
[20] D. Calegari and D. Gabai, ”Shrinkwrapping and the taming of hyperbolic 3-manifolds,” J. Amer. Math. Soc., vol. 19, iss. 2, pp. 385-446, 2006. · Zbl 1090.57010 · doi:10.1090/S0894-0347-05-00513-8 · arxiv:math/0407161
[21] R. D. Canary and D. McCullough, The refined relative compression body neighborhood, 2000. · www.math.ou.edu
[22] R. D. Canary and D. McCullough, ”Homotopy equivalences of 3-manifolds and deformation theory of Kleinian groups,” Mem. Amer. Math. Soc., vol. 172, iss. 812, p. xii, 2004. · Zbl 1062.57027
[23] R. D. Canary and Y. N. Minsky, ”On limits of tame hyperbolic \(3\)-manifolds,” J. Differential Geom., vol. 43, iss. 1, pp. 1-41, 1996. · Zbl 0856.57011 · projecteuclid.org
[24] R. D. Canary, ”Ends of hyperbolic \(3\)-manifolds,” J. Amer. Math. Soc., vol. 6, iss. 1, pp. 1-35, 1993. · Zbl 0810.57006 · doi:10.2307/2152793
[25] R. D. Canary, ”A covering theorem for hyperbolic \(3\)-manifolds and its applications,” Topology, vol. 35, iss. 3, pp. 751-778, 1996. · Zbl 0863.57010 · doi:10.1016/0040-9383(94)00055-7
[26] R. D. Canary, D. B. A. Epstein, and P. Green, ”Notes on notes of Thurston,” in Analytical and Geometric Aspects of Hyperbolic Space, Cambridge: Cambridge Univ. Press, 1987, vol. 111, pp. 3-92. · Zbl 0612.57009
[27] A. Douady and C. J. Earle, ”Conformally natural extension of homeomorphisms of the circle,” Acta Math., vol. 157, iss. 1-2, pp. 23-48, 1986. · Zbl 0615.30005 · doi:10.1007/BF02392590
[28] D. B. A. Epstein and A. Marden, ”Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces,” in Analytical and Geometric Aspects of Hyperbolic Space, Cambridge: Cambridge Univ. Press, 1987, vol. 111, pp. 113-253. · Zbl 0612.57010
[29] R. A. Evans, ”Tameness persists in weakly type-preserving strong limits,” Amer. J. Math., vol. 126, iss. 4, pp. 713-737, 2004. · Zbl 1058.57010 · doi:10.1353/ajm.2004.0028
[30] M. Freedman, J. Hass, and P. Scott, ”Least area incompressible surfaces in \(3\)-manifolds,” Invent. Math., vol. 71, iss. 3, pp. 609-642, 1983. · Zbl 0482.53045 · doi:10.1007/BF02095997 · eudml:143008
[31] U. Hamenstädt, ”Train tracks and the Gromov boundary of the complex of curves,” in Spaces of Kleinian Groups, Cambridge: Cambridge Univ. Press, 2006, vol. 329, pp. 187-207. · Zbl 1117.30036 · doi:10.1017/CBO9781139106993.009
[32] W. J. Harvey, ”Boundary structure of the modular group,” in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, Princeton, N.J.: Princeton Univ. Press, 1981, vol. 97, pp. 245-251. · Zbl 0461.30036
[33] W. J. Harvey, ”Modular groups and representation spaces,” in Geometry of Group Representations, Providence, RI: Amer. Math. Soc., 1988, vol. 74, pp. 205-214. · Zbl 0689.20038 · doi:10.1090/conm/074
[34] J. Hempel, \(3\)-Manifolds, Princeton, NJ: Princeton Univ. Press, 1976, vol. 86. · Zbl 0345.57001
[35] W. Jaco and H. J. Rubinstein, ”PL minimal surfaces in \(3\)-manifolds,” J. Differential Geom., vol. 27, iss. 3, pp. 493-524, 1988. · Zbl 0652.57005 · projecteuclid.org
[36] W. Jaco, Lectures on Three-Manifold Topology, Providence, R.I.: Amer. Math. Soc., 1980. · Zbl 0433.57001
[37] W. H. Jaco and P. B. Shalen, ”Seifert fibered spaces in \(3\)-manifolds,” Mem. Amer. Math. Soc., vol. 21, iss. 220, p. viii, 1979. · Zbl 0415.57005
[38] K. Johannson, Topology and Combinatorics of 3-Manifolds, New York: Springer-Verlag, 1995, vol. 1599. · Zbl 0820.57001 · doi:10.1007/BFb0074005
[39] T. Jørgensen and A. Marden, ”Algebraic and geometric convergence of Kleinian groups,” Math. Scand., vol. 66, iss. 1, pp. 47-72, 1990. · Zbl 0738.30032 · eudml:167093
[40] E. Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller space. · Zbl 1003.53053 · www.ericaklarreich.com
[41] I. Kra, ”On spaces of Kleinian groups,” Comment. Math. Helv., vol. 47, pp. 53-69, 1972. · Zbl 0239.30020 · doi:10.1007/BF02566788 · eudml:139499
[42] J. Luukkainen, ”Bi-Lipschitz concordance implies bi-Lipschitz isotopy,” Monatsh. Math., vol. 111, iss. 1, pp. 35-46, 1991. · Zbl 0719.30013 · doi:10.1007/BF01299275 · eudml:178508
[43] A. Marden, ”The geometry of finitely generated kleinian groups,” Ann. of Math., vol. 99, pp. 383-462, 1974. · Zbl 0282.30014 · doi:10.2307/1971059
[44] B. Maskit, ”Self-maps on Kleinian groups,” Amer. J. Math., vol. 93, pp. 840-856, 1971. · Zbl 0227.32007 · doi:10.2307/2373474
[45] H. A. Masur and Y. N. Minsky, ”Geometry of the complex of curves. I. Hyperbolicity,” Invent. Math., vol. 138, iss. 1, pp. 103-149, 1999. · Zbl 0941.32012 · doi:10.1007/s002220050343 · arxiv:math/9804098
[46] H. A. Masur and Y. N. Minsky, ”Geometry of the complex of curves. II. Hierarchical structure,” Geom. Funct. Anal., vol. 10, iss. 4, pp. 902-974, 2000. · Zbl 0972.32011 · doi:10.1007/PL00001643
[47] C. T. McMullen, ”Complex earthquakes and Teichmüller theory,” J. Amer. Math. Soc., vol. 11, iss. 2, pp. 283-320, 1998. · Zbl 0890.30031 · doi:10.1090/S0894-0347-98-00259-8
[48] Y. N. Minsky, ”On Thurston’s ending lamination conjecture,” in Low-Dimensional Topology, Int. Press, Cambridge, MA, 1994, vol. III, pp. 109-122. · Zbl 0846.57010
[49] Y. N. Minsky, ”The classification of punctured-torus groups,” Ann. of Math., vol. 149, iss. 2, pp. 559-626, 1999. · Zbl 0939.30034 · doi:10.2307/120976 · www.math.princeton.edu · eudml:120365 · arxiv:math/9807001
[50] Y. N. Minsky, ”Kleinian groups and the complex of curves,” Geom. Topol., vol. 4, pp. 117-148, 2000. · Zbl 0953.30027 · doi:10.2140/gt.2000.4.117 · emis:journals/UW/gt/GTVol4/paper3.abs.html · eudml:120728 · arxiv:math/9907070
[51] Y. N. Minsky, ”Bounded geometry for Kleinian groups,” Invent. Math., vol. 146, iss. 1, pp. 143-192, 2001. · Zbl 1061.37026 · doi:10.1007/s002220100163 · arxiv:math/0105078
[52] Y. N. Minsky, ”Combinatorial and geometrical aspects of hyperbolic 3-manifolds,” in Kleinian Groups and Hyperbolic 3-Manifolds, Komori, Y., Markovic, V., and Series, C., Eds., Cambridge: Cambridge Univ. Press, 2003, vol. 299, p. viii. · Zbl 1062.30053 · doi:10.1017/CBO9780511542817 · arxiv:math/0205173
[53] Y. N. Minsky, ”End invariants and the classification of hyperbolic 3-manifolds,” in Current Developments in Mathematics, 2002, Int. Press, Somerville, MA, 2003, pp. 181-217. · Zbl 1049.57010
[54] Y. N. Minsky, ”The classification of Kleinian surface groups. I. Models and bounds,” Ann. of Math., vol. 171, iss. 1, pp. 1-107, 2010. · Zbl 1193.30063 · doi:10.4007/annals.2010.171.1 · annals.princeton.edu · arxiv:math/0302208
[55] G. D. Mostow, ”Quasi-conformal mappings in \(n\)-space and the rigidity of hyperbolic space forms,” Inst. Hautes Études Sci. Publ. Math., vol. 34, pp. 53-104, 1968. · Zbl 0189.09402 · doi:10.1007/BF02684590 · numdam:PMIHES_1968__34__53_0 · eudml:103882
[56] H. Namazi, Heegaard Splittings and Hyperbolic Geometry, ProQuest LLC, Ann Arbor, MI, 2005. · gateway.proquest.com
[57] H. Namazi and J. Souto, Revisiting Thurston’s uniform injectivity theorem.
[58] H. Namazi and J. Souto, Non-realizability and ending laminations - Proof of the density conjecture, 2010. · Zbl 1258.57010 · doi:10.1007/s11511-012-0088-0
[59] K. Ohshika, Realising end invariants by limits of minimally parabolic, geometrically finite groups. · Zbl 1241.30014 · doi:10.2140/gt.2011.15.827 · arxiv:math/0504546
[60] K. Ohshika, ”Ending laminations and boundaries for deformation spaces of Kleinian groups,” J. London Math. Soc., vol. 42, iss. 1, pp. 111-121, 1990. · Zbl 0715.30032 · doi:10.1112/jlms/s2-42.1.111
[61] K. Ohshika, ”Constructing geometrically infinite groups on boundaries of deformation spaces,” J. Math. Soc. Japan, vol. 61, iss. 4, pp. 1261-1291, 2009. · Zbl 1195.57040 · doi:10.2969/jmsj/06141261 · www.jstage.jst.go.jp · arxiv:0809.1261
[62] K. Ohshika and H. Miyachi, ”On topologically tame Kleinian groups with bounded geometry,” in Spaces of Kleinian Groups, Cambridge: Cambridge Univ. Press, 2006, vol. 329, pp. 29-48. · Zbl 1104.57009
[63] J. Otal, ”Sur le nouage des géodésiques dans les variétés hyperboliques,” C. R. Acad. Sci. Paris Sér. I Math., vol. 320, iss. 7, pp. 847-852, 1995. · Zbl 0840.57008
[64] J. Otal, ”Les géodésiques fermées d’une variété hyperbolique en tant que nœuds,” in Kleinian Groups and Hyperbolic 3-Manifolds, Cambridge: Cambridge Univ. Press, 2003, vol. 299, pp. 95-104. · Zbl 1049.57007
[65] G. Prasad, ”Strong rigidity of \({\mathbf Q}\)-rank \(1\) lattices,” Invent. Math., vol. 21, pp. 255-286, 1973. · Zbl 0264.22009 · doi:10.1007/BF01418789 · eudml:142232
[66] M. Rees, The ending laminations theorem direct from Teichmüller geodesics, 2004.
[67] C. P. Rourke and B. J. Sanderson, Introduction to Piecewise-Linear Topology, New York: Springer-Verlag, 1972, vol. 69. · Zbl 0254.57010
[68] T. Soma, ”Function groups in Kleinian groups,” Math. Ann., vol. 292, iss. 1, pp. 181-190, 1992. · Zbl 0739.30033 · doi:10.1007/BF01444616 · eudml:164909
[69] T. Soma, Geometric approach to Ending Lamination Conjecture, 2008.
[70] J. Souto, Short curves in hyperbolic manifolds are not knotted, 2004.
[71] D. Sullivan, ”Hyperbolic geometry and homeomorphisms,” in Geometric Topology, New York: Academic Press, 1979, pp. 543-555. · Zbl 0478.57007
[72] D. Sullivan, ”On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions,” in Riemann Surfaces and Related Topics, Princeton, N.J.: Princeton Univ. Press, 1981, vol. 97, pp. 465-496. · Zbl 0567.58015
[73] D. Sullivan, ”Quasiconformal homeomorphisms and dynamics. II. Structural stability implies hyperbolicity for Kleinian groups,” Acta Math., vol. 155, iss. 3-4, pp. 243-260, 1985. · Zbl 0606.30044 · doi:10.1007/BF02392543
[74] P. Susskind, ”Kleinian groups with intersecting limit sets,” J. Analyse Math., vol. 52, pp. 26-38, 1989. · Zbl 0677.30028 · doi:10.1007/BF02820470
[75] W. P. Thurston, Hyperbolic structures on 3-manifolds, II: Surface groups and manifolds which fiber over the circle.
[76] W. P. Thurston, Hyperbolic structures on 3-manifolds, III: Deformations of 3-manifolds with incompressible boundary. · Zbl 0668.57015
[77] W. P. Thurston, The geometry and topology of 3-manifolds, 1982. · Zbl 0483.57007 · www.msri.org
[78] W. P. Thurston, ”Three-dimensional manifolds, Kleinian groups and hyperbolic geometry,” Bull. Amer. Math. Soc., vol. 6, iss. 3, pp. 357-381, 1982. · Zbl 0496.57005 · doi:10.1090/S0273-0979-1982-15003-0
[79] W. P. Thurston, ”Hyperbolic structures on \(3\)-manifolds. I: Deformation of acylindrical manifolds,” Ann. of Math., vol. 124, iss. 2, pp. 203-246, 1986. · Zbl 0668.57015 · doi:10.2307/1971277
[80] W. P. Thurston, Three-Dimensional Geometry and Topology. Vol. 1, Princeton, NJ: Princeton Univ. Press, 1997, vol. 35. · Zbl 0873.57001
[81] P. Tukia and J. Väisälä, ”Lipschitz and quasiconformal approximation and extension,” Ann. Acad. Sci. Fenn. Ser. A I Math., vol. 6, iss. 2, pp. 303-342 (1982), 1981. · Zbl 0448.30021
[82] J. Väisälä, ”Quasiconformal concordance,” Monatsh. Math., vol. 107, iss. 2, pp. 155-168, 1989. · Zbl 0685.30010 · doi:10.1007/BF01300921 · eudml:178419
[83] F. Waldhausen, ”On irreducible \(3\)-manifolds which are sufficiently large,” Ann. of Math., vol. 87, pp. 56-88, 1968. · Zbl 0157.30603 · doi:10.2307/1970594
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.