##
**The classification of Kleinian surface groups. I: Models and bounds.**
*(English)*
Zbl 1193.30063

This is the first of three papers dedicated to a proof of Thurston’s ending lamination conjecture: a hyperbolic 3-manifold with finitely generated fundamental group is uniquely determined by its topological type and its end invariants (Riemann surfaces or points in Teichmüller spaces for geometrically finite ends, ending laminations for geometrically infinite ones). If all ends are geometrically finite, the conjecture follows from the classical quasiconformal deformation theory of Kleinian groups of Ahlfors and Bers. When an end is geometrically infinite instead, there are two cases. If the boundary of the compact core of the manifold is incompressible, then work of Thurston and Bonahon gives a geometric and topological description of an end (which, in particular, is topologically tame, i.e., a product of a surface and an interval), and allows the ending lamination to be defined. If the core is compressible, the corresponding question of tameness of an end was resolved only recently by Agol and Calegari-Gabai (“the tameness conjecture”).

In the present paper, the incompressible boundary case is considered; by restriction to boundary subgroups, this reduces to the case of Kleinian surface groups. Very roughly, the idea is then to construct a model manifold depending only on the end invariants, together with a bilipschitz homeomorphism from the model to the manifold given by the Kleinian surface group. For two Kleinian surface groups with the same end invariants, this would result then in a bilipschitz homeomorphism between the two corresponding hyperbolic 3-manifolds which, by Sullivan’s rigidity theorem, could be deformed to an isometry. In the present paper, the model manifold together with a map satisfying some Lipschitz bounds is constructed. In the second paper of the series (together with Brock and Canary), this map will then be promoted to a bilipschitz homeomorphism, and finally, in the third paper, the case of a compact core with compressible boundary is considered.

A first version of the present paper has been submitted in 2003 and, as the author notes, in the meantime a number of other proofs of the ending lamination conjecture, including the compressible boundary case, have appeared, independently by Bowditch, Rees, and Soma. This finally resolves the last of the three major central conjectures in the theory of Kleinian groups, the tameness conjecture, the density conjecture, and the ending lamination conjecture, which had their origin in the classical theory of Kleinian groups as promoted by Ahlfors and Bers, and have been made accessible and finally resolved starting with the work of Thurston.

In the present paper, the incompressible boundary case is considered; by restriction to boundary subgroups, this reduces to the case of Kleinian surface groups. Very roughly, the idea is then to construct a model manifold depending only on the end invariants, together with a bilipschitz homeomorphism from the model to the manifold given by the Kleinian surface group. For two Kleinian surface groups with the same end invariants, this would result then in a bilipschitz homeomorphism between the two corresponding hyperbolic 3-manifolds which, by Sullivan’s rigidity theorem, could be deformed to an isometry. In the present paper, the model manifold together with a map satisfying some Lipschitz bounds is constructed. In the second paper of the series (together with Brock and Canary), this map will then be promoted to a bilipschitz homeomorphism, and finally, in the third paper, the case of a compact core with compressible boundary is considered.

A first version of the present paper has been submitted in 2003 and, as the author notes, in the meantime a number of other proofs of the ending lamination conjecture, including the compressible boundary case, have appeared, independently by Bowditch, Rees, and Soma. This finally resolves the last of the three major central conjectures in the theory of Kleinian groups, the tameness conjecture, the density conjecture, and the ending lamination conjecture, which had their origin in the classical theory of Kleinian groups as promoted by Ahlfors and Bers, and have been made accessible and finally resolved starting with the work of Thurston.

Reviewer: Bruno Zimmermann (Trieste)

### MSC:

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

57M50 | General geometric structures on low-dimensional manifolds |

### References:

[1] | W. Abikoff, ”Kleinian Groups-Geometrically Finite and Geometrically Perverse,” in Geometry of Group Representations (Boulder, CO, 1987), Providence, RI: Amer. Math. Soc., 1988, pp. 1-50. · Zbl 0662.30043 |

[2] | Agol, I., Tameness of hyperbolic 3-manifolds, preprint, 2004. |

[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 |

[4] | L. V. Ahlfors, ”Finitely generated Kleinian groups,” Amer. J. Math., vol. 86, pp. 413-429, 1964. · Zbl 0133.04201 |

[5] | J. M. Alonso and et al., ”Notes on word hyperbolic groups,” in Group Theory from a Geometrical Viewpoint (Trieste, 1990), World Sci. Publ., River Edge, NJ, 1991, pp. 3-63. · Zbl 0849.20023 |

[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 |

[7] | R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry, New York: Springer-Verlag, 1992. · Zbl 0768.51018 |

[8] | L. Bers, ”Simultaneous uniformization,” Bull. Amer. Math. Soc., vol. 66, pp. 94-97, 1960. · Zbl 0090.05101 |

[9] | L. Bers, ”On boundaries of Teichmüller spaces and on Kleinian groups. I,” Ann. of Math., vol. 91, pp. 570-600, 1970. · Zbl 0197.06001 |

[10] | L. Bers, ”Spaces of Kleinian groups,” in Several Complex Variables, (Proc. Conf., Univ. Maryland, College Park, Md., 1970), New York: Springer-Verlag, 1970, p. lecture notes in math. 155, 9-34. · Zbl 0211.10602 |

[11] | L. Bers, ”Spaces of degenerating Riemann surfaces,” in Discontinuous Groups and Riemann Surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), Princeton, NJ: Princeton Univ. Press, 1974, vol. 79, pp. 43-55. · Zbl 0294.32016 |

[12] | L. Bers, ”An inequality for Riemann surfaces,” in Differential Geometry and Complex Analysis, New York: Springer-Verlag, 1985, pp. 87-93. · Zbl 0575.30039 |

[13] | F. Bonahon, ”Bouts des variétés hyperboliques de dimension \(3\),” Ann. of Math., vol. 124, iss. 1, pp. 71-158, 1986. · Zbl 0671.57008 |

[14] | F. Bonahon, ”Geodesic laminations on surfaces,” in Laminations and Foliations in Dynamics, Geometry and Topology (Stony Brook, NY, 1998), Providence, RI: Amer. Math. Soc., 2001, vol. 269, pp. 1-37. · Zbl 0996.53029 |

[15] | B. H. Bowditch, ”Notes on Gromov’s hyperbolicity criterion for path-metric spaces,” in Group Theory from a Geometrical Viewpoint (Trieste, 1990), World Sci. Publ., River Edge, NJ, 1991, pp. 64-167. · Zbl 0843.20031 |

[16] | Bowditch, B. H., End invariants of hyperbolic 3-manifolds, 2005. |

[17] | J. F. Brock, ”Continuity of Thurston’s length function,” Geom. Funct. Anal., vol. 10, iss. 4, pp. 741-797, 2000. · Zbl 0968.57011 |

[18] | Brooks, Robert and Canary, Richard D and Minsky, Yair N., The classification of Kleinian surface groups II: the ending lamination conjecture, preprint. · Zbl 1253.57009 |

[19] | R. Brooks and P. J. Matelski, ”Collars in Kleinian groups,” Duke Math. J., vol. 49, iss. 1, pp. 163-182, 1982. · Zbl 0484.30029 |

[20] | P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Boston, MA: Birkhäuser, 1992. · Zbl 0770.53001 |

[21] | 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 |

[22] | R. D. Canary, ”Ends of hyperbolic \(3\)-manifolds,” J. Amer. Math. Soc., vol. 6, iss. 1, pp. 1-35, 1993. · Zbl 0810.57006 |

[23] | 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, 1984), Cambridge: Cambridge Univ. Press, 1987, pp. 3-92. · Zbl 0612.57009 |

[24] | J. W. Cannon, ”The theory of negatively curved spaces and groups,” in Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces (Trieste, 1989), New York: Oxford Univ. Press, 1991, pp. 315-369. · Zbl 0764.57002 |

[25] | A. J. Casson and S. A. Bleiler, Automorphisms of Surfaces after Nielsen and Thurston, Cambridge: Cambridge Univ. Press, 1988. · Zbl 0649.57008 |

[26] | 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 (Coventry/Durham, 1984), Cambridge: Cambridge Univ. Press, 1987, pp. 113-253. · Zbl 0612.57010 |

[27] | E. Ghys and P. de la Harpe (rm eds.), Sur les Groupes Hyperboliques d’après Mikhael Gromov, Boston, MA: Birkhäuser, 1990. · Zbl 0731.20025 |

[28] | G. Gromov, Essays in Group Theory, New York: Springer-Verlag, 1987. · Zbl 0547.00016 |

[29] | W. J. Harvey, ”Boundary structure of the modular group,” in Riemann Surfaces and Related Topics: Proc. 1978 Stony Brook Conference (State Univ. New York, Stony Brook, NY, 1978), Princeton, N.J.: Princeton Univ. Press, 1981, pp. 245-251. · Zbl 0461.30036 |

[30] | A. Hatcher and W. Thurston, ”A presentation for the mapping class group of a closed orientable surface,” Topology, vol. 19, iss. 3, pp. 221-237, 1980. · Zbl 0447.57005 |

[31] | S. Hersonsky, ”A generalization of the Shimizu-Leutbecher and Jørgensen inequalities to Möbius transformations in \({\mathbf R}^N\),” Proc. Amer. Math. Soc., vol. 121, iss. 1, pp. 209-215, 1994. · Zbl 0812.30017 |

[32] | T. Jørgensen, ”On discrete groups of Möbius transformations,” Amer. J. Math., vol. 98, iss. 3, pp. 739-749, 1976. · Zbl 0336.30007 |

[33] | D. A. Kazhdan and G. A. Margulis, ”A proof of Selberg’s conjecture,” Math. USSR Sb., vol. 4, pp. 147-152, 1968. · Zbl 0241.22024 |

[34] | Klarreich, E., The boundary at infinity of the curve complex and the relative Teichmüller space, preprint. · Zbl 1003.53053 |

[35] | I. Kra, ”On spaces of Kleinian groups,” Comment. Math. Helv., vol. 47, pp. 53-69, 1972. · Zbl 0239.30020 |

[36] | R. S. Kulkarni and P. B. Shalen, ”On Ahlfors’ finiteness theorem,” Adv. Math., vol. 76, iss. 2, pp. 155-169, 1989. · Zbl 0684.57019 |

[37] | A. Marden, ”The geometry of finitely generated kleinian groups,” Ann. of Math., vol. 99, pp. 383-462, 1974. · Zbl 0282.30014 |

[38] | A. Marden and B. Maskit, ”On the isomorphism theorem for Kleinian groups,” Invent. Math., vol. 51, iss. 1, pp. 9-14, 1979. · Zbl 0399.30037 |

[39] | B. Maskit, ”Self-maps on Kleinian groups,” Amer. J. Math., vol. 93, pp. 840-856, 1971. · Zbl 0227.32007 |

[40] | B. Maskit, Kleinian Groups, New York: Springer-Verlag, 1988. · Zbl 0627.30039 |

[41] | 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 |

[42] | 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 |

[43] | D. McCullough, ”Compact submanifolds of \(3\)-manifolds with boundary,” Quart. J. Math. Oxford Ser., vol. 37, iss. 147, pp. 299-307, 1986. · Zbl 0628.57008 |

[44] | R. Meyerhoff, ”A lower bound for the volume of hyperbolic \(3\)-manifolds,” Canad. J. Math., vol. 39, iss. 5, pp. 1038-1056, 1987. · Zbl 0694.57005 |

[45] | Y. N. Minsky, ”A geometric approach to the complex of curves on a surface,” in Topology and Teichmüller Spaces (Katinkulta, 1995), World Sci. Publ., River Edge, NJ, 1996, pp. 149-158. · Zbl 0937.30027 |

[46] | Y. N. Minsky, ”The classification of punctured-torus groups,” Ann. of Math., vol. 149, iss. 2, pp. 559-626, 1999. · Zbl 0939.30034 |

[47] | Y. N. Minsky, ”Kleinian groups and the complex of curves,” Geom. Topol., vol. 4, pp. 117-148, 2000. · Zbl 0953.30027 |

[48] | Y. N. Minsky, ”Bounded geometry for Kleinian groups,” Invent. Math., vol. 146, iss. 1, pp. 143-192, 2001. · Zbl 1061.37026 |

[49] | Kleinian Groups and Hyperbolic 3-ManifoldsCambridge: Cambridge Univ. Press, 2003. · Zbl 1031.30002 |

[50] | R. L. Moore, ”On the foundations of plane analysis situs,” Trans. Amer. Math. Soc., vol. 17, iss. 2, pp. 131-164, 1916. |

[51] | R. L. Moore, ”Concerning upper semi-continuous collections of continua,” Trans. Amer. Math. Soc., vol. 27, iss. 4, pp. 416-428, 1925. · JFM 51.0464.03 |

[52] | G. D. Mostow, ”Quasi-conformal mappings in \(n\)-space and the rigidity of hyperbolic space forms,” Inst. Hautes Études Sci. Publ. Math., iss. 34, pp. 53-104, 1968. · Zbl 0189.09402 |

[53] | R. C. Penner and J. L. Harer, Combinatorics of Train Tracks, Princeton, NJ: Princeton Univ. Press, 1992. · Zbl 0765.57001 |

[54] | G. Prasad, ”Strong rigidity of \({\mathbf Q}\)-rank \(1\) lattices,” Invent. Math., vol. 21, pp. 255-286, 1973. · Zbl 0264.22009 |

[55] | M. Rees, The ending laminations theorem direct from Teichmüller geodesics, 2004. |

[56] | G. P. Scott, ”Compact submanifolds of \(3\)-manifolds,” J. London Math. Soc., vol. 7, pp. 246-250, 1973. · Zbl 0266.57001 |

[57] | Soma, T., Geometric approach to Ending Lamination Conjecture, 2008, \marginparCheck. |

[58] | D. Sullivan, ”On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions,” in Riemann Surfaces and Related Topics: Proc. 1978 Stony Brook Conference (State Univ. New York, Stony Brook, NY, 1978), Princeton, NJ: Princeton Univ. Press, 1981, pp. 465-496. · Zbl 0567.58015 |

[59] | Thurston, William P., Hyperbolic structures on 3-manifolds, II: surface groups and manifolds which fiber over the circle, preprint. |

[60] | Thurston, William P., The geometry and topology of 3-manifolds, 1982. · Zbl 0483.57007 |

[61] | 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 |

[62] | 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 |

[63] | W. P. Thurston, Three-Dimensional Geometry and Topology. Vol. 1, Princeton, NJ: Princeton Univ. Press, 1997. · Zbl 0873.57001 |

[64] | P. L. Waterman, ”Möbius transformations in several dimensions,” Adv. Math., vol. 101, iss. 1, pp. 87-113, 1993. · Zbl 0793.15019 |

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.