Kleinian groups and the complex of curves.

*(English)*Zbl 0953.30027Summary: We examine the internal geometry of a Kleinian surface group and its relations to the asymptotic geometry of its ends, using the combinatorial structure of the complex of curves on the surface. Our main results give necessary conditions for the Kleinian group to have ‘bounded geometry’ (lower bounds on injectivity radius) in terms of a sequence of coefficients (subsurface projections) computed using the ending invariants of the group and the complex of curves. These results are directly analogous to those obtained in the case of punctured-torus surface groups. In that setting the ending invariants are points in the closed unit disk and the coefficients are closely related to classical continued-fraction coefficients. The estimates obtained play an essential role in the solution of Thurston’s ending lamination conjecture in that case.

##### MSC:

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

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

##### Keywords:

ending lamination; complex of curves; pleated surface; bounded geometry; injectivity radius##### References:

[1] | L Ahlfors, L Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. \((2)\) 72 (1960) 385 · Zbl 0104.29902 |

[2] | R Benedetti, C Petronio, Lectures on hyperbolic geometry, Universitext, Springer (1992) · Zbl 0768.51018 |

[3] | L Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960) 94 · Zbl 0090.05101 |

[4] | L Bers, On boundaries of Teichmüller spaces and on Kleinian groups I, Ann. of Math. \((2)\) 91 (1970) 570 · Zbl 0197.06001 |

[5] | F Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. \((2)\) 124 (1986) 71 · Zbl 0671.57008 |

[6] | F Bonahon, J P Otal, Variétés hyperboliques à géodésiques arbitrairement courtes, Bull. London Math. Soc. 20 (1988) 255 · Zbl 0648.53027 |

[7] | J F Brock, Continuity of Thurston’s length function, Geom. Funct. Anal. 10 (2000) 741 · Zbl 0968.57011 |

[8] | R D Canary, Algebraic convergence of Schottky groups, Trans. Amer. Math. Soc. 337 (1993) 235 · Zbl 0772.30037 |

[9] | R D Canary, A covering theorem for hyperbolic 3-manifolds and its applications, Topology 35 (1996) 751 · Zbl 0863.57010 |

[10] | R D Canary, D B A Epstein, P Green, Notes on notes of Thurston, London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press (1987) 3 · Zbl 0612.57009 |

[11] | A J Casson, S A Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts 9, Cambridge University Press (1988) · Zbl 0649.57008 |

[12] | A Fathi, F Laudenbach, V Poenaru, Travaux de Thurston sur les surfaces, Astérisque 66, Société Mathématique de France (1979) 284 |

[13] | W J Harvey, Boundary structure of the modular group, Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 245 · Zbl 0461.30036 |

[14] | A Hatcher, On triangulations of surfaces, Topology Appl. 40 (1991) 189 · Zbl 0727.57012 |

[15] | A E Hatcher, Measured lamination spaces for surfaces, from the topological viewpoint, Topology Appl. 30 (1988) 63 · Zbl 0662.57005 |

[16] | A Hatcher, W Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980) 221 · Zbl 0447.57005 |

[17] | J Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (2001) 631 · Zbl 0985.57014 |

[18] | H A Masur, Y N Minsky, Geometry of the complex of curves II: Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902 · Zbl 0972.32011 |

[19] | H A Masur, Y N Minsky, Geometry of the complex of curves I: Hyperbolicity, Invent. Math. 138 (1999) 103 · Zbl 0941.32012 |

[20] | C T McMullen, Renormalization and 3-manifolds which fiber over the circle, Annals of Mathematics Studies 142, Princeton University Press (1996) · Zbl 0860.58002 |

[21] | Y N Minsky, Harmonic maps into hyperbolic 3-manifolds, Trans. Amer. Math. Soc. 332 (1992) 607 · Zbl 0762.53040 |

[22] | Y N Minsky, On Thurston’s ending lamination conjecture, Conf. Proc. Lecture Notes Geom. Topology, III, Int. Press, Cambridge, MA (1994) 109 · Zbl 0846.57010 |

[23] | Y N Minsky, A geometric approach to the complex of curves on a surfaceuller spaces (Katinkulta, 1995)”, World Sci. Publ., River Edge, NJ (1996) 149 · Zbl 0937.30027 |

[24] | Y N Minsky, The classification of punctured-torus groups, Ann. of Math. \((2)\) 149 (1999) 559 · Zbl 0939.30034 |

[25] | K Ohshika, Ending laminations and boundaries for deformation spaces of Kleinian groups, J. London Math. Soc. \((2)\) 42 (1990) 111 · Zbl 0715.30032 |

[26] | R C Penner, J L Harer, Combinatorics of train tracks, Annals of Mathematics Studies 125, Princeton University Press (1992) · Zbl 0765.57001 |

[27] | W P Thurston, Hyperbolic structures on 3-manifolds II: surface groups and manifolds which fiber over the circle, |

[28] | W P Thurston, Hyperbolic structures on 3-manifolds III: deformations of 3-manifolds with incompressible boundary, |

[29] | W P Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series 35, Princeton University Press (1997) · Zbl 0873.57001 |

[30] | W P Thurston, Hyperbolic structures on 3-manifolds I: Deformation of acylindrical manifolds, Ann. of Math. \((2)\) 124 (1986) 203 · Zbl 0668.57015 |

[31] | W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. \((\)N.S.\()\) 19 (1988) 417 · Zbl 0674.57008 |

[32] | W P Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series 35, Princeton University Press (1997) · Zbl 0873.57001 |

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.