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

Minsky, Yair N.

Journal of the American Mathematical Society, Tome 07 (1994) no. 3, pp. 539-588

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-1994-1257060-3

Bibliographie

[1] Ahlfors, Lars, Bers, Lipman Riemann’s mapping theorem for variable metrics Ann. of Math. (2) 1960 385 404

[2] Abikoff, William The real analytic theory of Teichmüller space 1980

[3] Abikoff, William Kleinian groups—geometrically finite and geometrically perverse 1988 1 50

[4] Ahlfors, Lars V. Conformal invariants: topics in geometric function theory 1973

[5] Beardon, Alan F. The geometry of discrete groups 1983

[6] Bers, Lipman Simultaneous uniformization Bull. Amer. Math. Soc. 1960 94 97

[7] Bers, Lipman Spaces of Kleinian groups 1970 9 34

[8] Bonahon, Francis Bouts des variétés hyperboliques de dimension 3 Ann. of Math. (2) 1986 71 158

[9] Bonahon, Francis Bouts des variétés hyperboliques de dimension 3 Ann. of Math. (2) 1986 71 158

[10] Cannon, James W. The theory of negatively curved spaces and groups 1991 315 369

[11] Canary, Richard D. Ends of hyperbolic 3-manifolds J. Amer. Math. Soc. 1993 1 35

[12] Casson, Andrew J., Bleiler, Steven A. Automorphisms of surfaces after Nielsen and Thurston 1988

[13] Coornaert, M., Delzant, T., Papadopoulos, A. Géométrie et théorie des groupes 1990

[14] Canary, R. D., Epstein, D. B. A., Green, P. Notes on notes of Thurston 1987 3 92

[15] Cannon, James W., Thurston, William P. Group invariant Peano curves Geom. Topol. 2007 1315 1355

[16] Epstein, D. B. A., Marden, A. Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces 1987 113 253

[17] Fenley, Sérgio R. Asymptotic properties of depth one foliations in hyperbolic 3-manifolds J. Differential Geom. 1992 269 313

[18] Floyd, William J. Group completions and limit sets of Kleinian groups Invent. Math. 1980 205 218

[19] Gardiner, Frederick P. Teichmüller theory and quadratic differentials 1987

[20] Gardiner, Frederick P., Masur, Howard Extremal length geometry of Teichmüller space Complex Variables Theory Appl. 1991 209 237

[21] Gromov, M. Hyperbolic groups 1987 75 263

[22] Hatcher, A. E. Measured lamination spaces for surfaces, from the topological viewpoint Topology Appl. 1988 63 88

[23] Hubbard, John, Masur, Howard Quadratic differentials and foliations Acta Math. 1979 221 274

[24] Kerckhoff, Steven P. The asymptotic geometry of Teichmüller space Topology 1980 23 41

[25] Kerckhoff, Steven P. Lines of minima in Teichmüller space Duke Math. J. 1992 187 213

[26] Kra, Irwin On spaces of Kleinian groups Comment. Math. Helv. 1972 53 69

[27] Levitt, Gilbert Foliations and laminations on hyperbolic surfaces Topology 1983 119 135

[28] Marden, Albert The geometry of finitely generated kleinian groups Ann. of Math. (2) 1974 383 462

[29] Masur, Howard Hausdorff dimension of the set of nonergodic foliations of a quadratic differential Duke Math. J. 1992 387 442

[30] Maskit, Bernard On boundaries of Teichmüller spaces and on Kleinian groups. II Ann. of Math. (2) 1970 607 639

[31] Maskit, Bernard Self-maps on Kleinian groups Amer. J. Math. 1971 840 856

[32] Masur, Howard Uniquely ergodic quadratic differentials Comment. Math. Helv. 1980 255 266

[33] Masur, Howard Two boundaries of Teichmüller space Duke Math. J. 1982 183 190

[34] Maskit, Bernard Comparison of hyperbolic and extremal lengths Ann. Acad. Sci. Fenn. Ser. A I Math. 1985 381 386

[35] Mcmullen, Curt Amenability, Poincaré series and quasiconformal maps Invent. Math. 1989 95 127

[36] Minsky, Yair N. Harmonic maps into hyperbolic 3-manifolds Trans. Amer. Math. Soc. 1992 607 632

[37] Minsky, Yair N. Teichmüller geodesics and ends of hyperbolic 3-manifolds Topology 1993 625 647

[38] Mostow, G. D. Quasi-conformal mappings in 𝑛-space and the rigidity of hyperbolic space forms Inst. Hautes Études Sci. Publ. Math. 1968 53 104

[39] Mostow, G. D. Strong rigidity of locally symmetric spaces 1973

[40] Prasad, Gopal Strong rigidity of 𝑄-rank 1 lattices Invent. Math. 1973 255 286

[41] Scott, G. P. Compact submanifolds of 3-manifolds J. London Math. Soc. (2) 1973 246 250

[42] Strebel, Kurt Quadratic differentials 1984

[43] Sullivan, Dennis On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions 1981 465 496

[44] Thurston, William P. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry Bull. Amer. Math. Soc. (N.S.) 1982 357 381

[45] Thurston, William P. Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds Ann. of Math. (2) 1986 203 246