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On rigidity, limit sets, and end invariants of hyperbolic $ 3$-manifolds


Author: Yair N. Minsky
Journal: J. Amer. Math. Soc. 7 (1994), 539-588
MSC: Primary 57M50; Secondary 30C62, 30F40, 57M60, 57N10
DOI: https://doi.org/10.1090/S0894-0347-1994-1257060-3
MathSciNet review: 1257060
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Abstract: Thurston's ending lamination conjecture states that a hyperbolic manifold is uniquely determined by a collection of Riemann surfaces and geodesic laminations that describe the asymptotic geometry of its ends. We prove this conjecture for the case of manifolds whose fundamental group is freely indecomposable, and which admit a positive lower bound on injectivity radii.

The techniques of the proof apply to show that a Kleinian surface group admitting a positive lower bound on injectivity radii is continuously semiconjugate to a Fuchsian group. This extends results of Cannon and Thurston.

A further consequence is a rigidity theorem for surface groups satisfying the injectivity radius condition, namely that two such groups whose actions on the sphere are conjugate by a homeomorphism that is conformal on the domains of discontinuity must be conjugate by a Möbius transformation.


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  • [AB60] L. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics., Ann. of Math. (2) 72 (1960), 385-404. MR 0115006 (22:5813)
  • [Abi80] W. Abikoff, The real-analytic theory of Teichmüller space, Lecture Notes in Math., vol. 820, Springer-Verlag, Berlin and New York, 1980. MR 590044 (82a:32028)
  • [Abi88] -, Kleinian groups - geometrically finite and geometrically perverse, Geometry of Group Representations, Contemporary Math., no. 74, Amer. Math. Soc., Providence, RI, 1988, pp. 11-50. MR 957510 (89j:30061)
  • [Ahl73] L. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill, New York, 1973. MR 0357743 (50:10211)
  • [Bea83] A. F. Beardon, The geometry of discrete groups, Springer-Verlag, Berlin and New York, 1983. MR 698777 (85d:22026)
  • [Ber60] L. Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94-97. MR 0111834 (22:2694)
  • [Ber70] -, Spaces of Kleinian groups, Maryland Conference in Several Complex Variables I, Lecture Notes in Math, vol. 155, Springer-Verlag, Berlin and New York, 1970, pp. 9-34. MR 0271333 (42:6216)
  • [Bon] F. Bonahon, Bouts des variétés hyperboliques de dimension 3, prepublications de Orsay, 1985. MR 847953 (88c:57013)
  • [Bon86] -, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. (2) 124 (1986), 71-158. MR 847953 (88c:57013)
  • [Can91] J. Cannon, The theory of negatively curved spaces and groups, Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces (Trieste, 1989), Oxford Univ. Press, London and New York, 1991, pp. 315-369. MR 1130181
  • [Can93] R. D. Canary, Ends of hyperbolic $ 3$-manifolds, J. Amer. Math. Soc. 6 (1993), 1-35. MR 1166330 (93e:57019)
  • [Cas82] A. J. Casson, Automorphisms of surfaces after Nielsen and Thurston, Notes by S. A. Bleiler, U. T. Austin, 1982.
  • [CB88] A. J. Casson and S. A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, Cambridge Univ. Press, London and New York, 1988. MR 964685 (89k:57025)
  • [CDP90] M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et theorie de groupes: les groups hyperboliques de Gromov, Springer-Verlag, Berlin and New York, 1990. MR 1075994 (92f:57003)
  • [CEG87] R. D. Canary, D. B. A. Epstein, and P. Green, Notes on notes of Thurston, Analytical and Geometric Aspects of Hyperbolic Space, London Math. Soc. Lecture Note Ser., no. 111, Cambridge Univ. Press, Cambridge and New York, 1987, pp. 3-92. MR 903850 (89e:57008)
  • [CT89] J. Cannon and W. Thurston, Group invariant peano curves, preprint, 1989. MR 2326947 (2008i:57016)
  • [EM87] D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, Analytical and Geometric Aspects of Hyperbolic Space, London Math. Soc. Lecture Note Ser., no. 111, Cambridge Univ. Press, Cambridge and New York, 1987, pp. 113-254. MR 903852 (89c:52014)
  • [Fen92] S. Fenley, Asymptotic properties of depth one foliations in hyperbolic $ 3$-manifolds, J. Differential Geom. 36 (1992), 269-313. MR 1180384 (93k:57030)
  • [Flo80] W. J. Floyd, Group completions and limit sets of Kleinian groups., Invent. Math. 57 (1980), 205-218. MR 568933 (81e:57002)
  • [FLP79] A. Fathi, F. Laudenbach, and V. Poenaru, Travaux de Thurston sur les surfaces, vol. 66-67, Asterisque, 1979.
  • [Gar87] F. Gardiner, Teichmüller theory and quadratic differentials, Wiley Interscience, New York, 1987. MR 903027 (88m:32044)
  • [GdlH90] E. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d'aprés Mikhael Gromov, Birkhaüser, Basel, 1990.
  • [GM91] F. Gardiner and H. Masur, Extremal length geometry of Teichmüller space, Complex Variables Theory Appl. 16 (1991), 209-237. MR 1099913 (92f:32034)
  • [Gro87] M. Gromov, Hyperbolic groups, Essays in Group Theory (S. M. Gersten, ed.), Math. Sci. Res. Inst. Publ., no. 8, Springer-Verlag, Berlin and New York, 1987. MR 919829 (89e:20070)
  • [Hat88] A. E. Hatcher, Measured lamination spaces for surfaces, from the topological viewpoint, Topology Appl. 30 (1988), 63-88. MR 964063 (89k:57022)
  • [HM79] J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), 221-274. MR 523212 (80h:30047)
  • [Ker80] S. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980), 23-41. MR 559474 (81f:32029)
  • [Ker92] -, Lines of minima in Teichmüller space, Duke Math. J. 65 (1992), 187-213. MR 1150583 (93b:32027)
  • [Kra72] I. Kra, On spaces of Kleinian groups, Comment. Math. Helv. 47 (1972), 53-69. MR 0306485 (46:5611)
  • [Lev83] G. Levitt, Foliations and laminations on hyperbolic surfaces, Topology 22 (1983), 119-135. MR 683752 (84h:57015)
  • [Mar74] A. Marden, The geometry of finitely generated Kleinian groups, Ann. of Math. (2) 99 (1974), 383-462. MR 0349992 (50:2485)
  • [Mas] H. Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, preprint. MR 1167101 (93f:30045)
  • [Mas70] B. Maskit, On boundaries of Teichmüller spaces and on Kleinian groups II, Ann. of Math. (2) 91 (1970), 607-639. MR 0297993 (45:7045)
  • [Mas71] -, Self-maps of Kleinian groups, Amer. J. Math. 93 (1971), 840-856. MR 0291453 (45:544)
  • [Mas80] H. Masur, Uniquely ergodic quadratic differentials, Comment. Math. Helv. 55 (1980), 255-266. MR 576605 (82a:32031)
  • [Mas82] -, Two boundaries of Teichmüller space, Duke Math. J. 49 (1982), 183-190. MR 650376 (83k:32035)
  • [Mas85] B. Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. 10 (1985), 381-386. MR 802500 (87c:30062)
  • [McM89] C. McMullen, Amenability, Poincaré series and quasiconformal maps, Invent. Math. 97 (1989), 95-127. MR 999314 (90e:30048)
  • [Min92] Y. Minsky, Harmonic maps into hyperbolic $ 3$-manifolds, Trans. Amer. Math. Soc. 332 (1992), 607-632. MR 1100698 (92j:58027)
  • [Min93] -, Teichmüller geodesics and ends of hyperbolic $ 3$-manifolds, Topology 32 (1993), 625-647. MR 1231968 (95g:57031)
  • [Mos68] G. D. Mostow, Quasiconformal mappings in $ n$-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53-104. MR 0236383 (38:4679)
  • [Mos73] -, Strong rigidity of locally symmetric spaces, Ann. of Math. Stud., no. 78, Princeton Univ. Press, Princeton, NJ, 1973. MR 0385004 (52:5874)
  • [Ohs] K. Ohshika, Topologically conjugate Kleinian groups, preprint.
  • [Pra73] G. Prasad, Strong rigidity of $ Q$-rank 1 lattices, Invent. Math. 21 (1973), 255-286. MR 0385005 (52:5875)
  • [Sco73] P. Scott, Compact submanifolds of $ 3$-manifolds, J. London Math. Soc. 7 (1973), 246-250. MR 0326737 (48:5080)
  • [Str84] K. Strebel, Quadratic differentials, Springer-Verlag, Berlin and New York, 1984. MR 743423 (86a:30072)
  • [Sul81] D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, Ann. of Math. Stud., no. 97, Princeton Univ. Press, Princeton, NJ, 1981. MR 624833 (83f:58052)
  • [Thua] W. Thurston, Hyperbolic structures on $ 3$-manifolds, II: surface groups and manifolds which fiber over the circle, preprint.
  • [Thub] -, Minimal stretch maps between hyperbolic surfaces, unpublished manuscript, Princeton University.
  • [Thu82a] -, The geometry and topology of $ 3$-manifolds, Princeton University lecture notes, 1982.
  • [Thu82b] -, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357-381. MR 648524 (83h:57019)
  • [Thu86] -, Hyperbolic structures on $ 3$-manifolds, I: deformation of acylindrical manifolds, Ann. of Math. (2) 124 (1986), 203-246. MR 855294 (88g:57014)

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DOI: https://doi.org/10.1090/S0894-0347-1994-1257060-3
Article copyright: © Copyright 1994 American Mathematical Society

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