Acta Mathematica

Hausdorff dimension and Kleinian groups

Christopher J. Bishop and Peter W. Jones

Full-text: Open access

Note

The first author is partially supported by NSF Grant DMS-92-04092 and an Alfred P. Sloan research fellowship. The second author is partially supported by NSF Grant DMS-92-13595.

Article information

Source
Acta Math., Volume 179, Number 1 (1997), 1-39.

Dates
Received: 29 November 1994
Revised: 28 January 1997
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485891069

Digital Object Identifier
doi:10.1007/BF02392718

Mathematical Reviews number (MathSciNet)
MR1484767

Zentralblatt MATH identifier
0921.30032

Rights
1997 © Institut Mittag-Leffler

Citation

Bishop, Christopher J.; Jones, Peter W. Hausdorff dimension and Kleinian groups. Acta Math. 179 (1997), no. 1, 1--39. doi:10.1007/BF02392718. https://projecteuclid.org/euclid.acta/1485891069


Export citation

References

  • Abikoff, W., On boundaries of Teichmüller spaces and on Kleinian groups, III. Acta Math., 134 (1976), 212–237.
  • Ahlfors, L., Finitely generated Kleinian groups. Amer. J. Math., 86 (1964), 413–429.
  • Aravinda, C. S., Bounded geodesics and Hausdorff dimension. Math. Proc. Cambridge Philos. Soc., 116 (1994), 505–511.
  • Aravinda, C. S. & Leuzinger, E., Bounded geodesics in rank-1 locally symmetric spaces. Ergodic Theory Dynamical Systems, 15 (1995), 813–820.
  • Beardon, A. F., The Geometry of Discrete Groups. Graduate texts in Math., 91, Springer-Verlag, New York-Berlin, 1983.
  • Beardon, A. F. & Maskit, B., Limit points of Kleinian groups and finite sided fundamental polyhedra. Acta Math., 132 (1974), 1–12.
  • Benedetti, R. & Petronio, C., Lectures on Hyperbolic Geometry. Universitext, Springer-Verlag, Berlin, 1992.
  • Bers, L., Inequalities for finitely generated Kleinian groups. J. Analyse Math., 18 (1967), 23–41.
  • — On the boundaries of Teichmüller spaces and on Kleinian groups, I. Ann. of Math., 91 (1970), 570–600.
  • Bishop, C. J., Minkowski dimension and the Poincaré exponent. Michigan Math. J., 43 (1996), 231–246.
  • — Geometric exponents and Kleinian groups. Invent. Math., 127 (1997), 33–50.
  • Bishop, C. J. & Jones, P. W., Wiggly sets and limit sets. Ark. Mat., 35 (1997), 201–224.
  • — The law of the iterated logarithim for Kleinian groups, to appear in Lipa's Legacy (J. Dodziuk and L. Keen, eds.). Contemp. Math., Amer. Math. Soc., Providence, RI, 1997.
  • Bowditch, B. H., Geometrical finiteness for hyperbolic groups. J. Funct. Anal., 113 (1993), 245–317.
  • Bowen, R., Hausdorff dimension of quasicircles. Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11–25.
  • Braam, P., A Kaluza-Klein approach to hyperbolic three-manifolds. Enseign. Math., 34 (1988), 275–311.
  • Bullet, S. & Mantica, G., Group theory of hyperbolic circle packings. Nonlinearity, 5 (1992), 1085–1109.
  • Canary, R. D., The Poincaré metric and a conformal version of a theorem of Thurston. Duke Math. J., 64 (1991), 349–359.
  • — On the Laplacian and geometry of hyperbolic 3-manifolds. J. Differential Geom., 36 (1992), 349–367.
  • — Ends of hyperbolic 3-manifolds. J. Amer. Math. Soc., 6 (1993), 1–35.
  • Canary, R. D. & Taylor, E., Kleinian groups with small limit sets. Duke Math. J., 73 (1994), 371–381.
  • Carleson, L., Selected Problems on Exceptional Sets. Van Nostrand Math. Stud., 13. Van Nostrand, Princeton, NJ-Toronto, ON-London, 1967.
  • Chavel, I., Eigenvalues in Riemannian Geometry. Pure Appl. Math., 115. Academic Press, Orlando, FL, 1984.
  • Cheeger, J. & Ebin, D. G., Comparison Theorems in Riemannian Geometry. North-Holland Math. Libary, 9. North-Holland, Amsterdam-Oxford, 1975.
  • Corlette, K., Hausdorff dimensions of limit sets, I. Invent Math., 102 (1990), 521–542.
  • Corlette, K. & Iozzi, A., Limit sets of isometry groups of exotic hyperbolic spaces. Preprint, 1994.
  • Dani, S. G., Bounded orbits of flows on homogeneous spaces. Comment. Math. Helv., 61 (1986), 636–660.
  • Davies, E. B., Gaussian upper bounds for the heat kernel of some second-order operators on Riemannian manifolds. J. Funct. Anal., 80 (1988), 16–32.
  • Heat Kernels and Spectral Theory. Cambridge Tracts in Math., 92. Cambridge Univ. Press, Cambridge-New York, 1989.
  • — The state of the art for heat kernel bounds on negatively curved manifolds. Bull. London Math. Soc., 25 (1993), 289–292.
  • Donnelly, H., Essential spectrum and heat kernel. J. Funct. Anal., 75 (1987), 362–381.
  • Epstein, D. B. A. & Marden, A., Convex hulls in hyperbolic spaces, a theorem of Sullivan and measured pleated surfaces, in Analytical and Geometric Aspects of Hyperbolic Space, pp. 113–253. London Math. Soc. Lecture Note Ser., 111. Cambridge Univ. Press, Cambridge-New York, 1987.
  • Fernández, J. L., Domains with strong barrier. Rev. Mat. Iberoamericana, 5 (1989), 47–65.
  • Fernández, J. L. & Melián, M. V., Bounded geodesics of Riemann surfaces and hyperbolic manifolds. Trans. Amer. Math. Soc., 347 (1995), 3533–3549.
  • González, M. J., Uniformly perfect sets, Green's function and fundamental domains. Rev. Mat. Iberoamericana, 8 (1992), 239–269.
  • Greenberg, L., Fundamental polyhedron for Kleinian groups. Ann. of Math., 84 (1966), 433–441.
  • Grigor'yan, A., Heat kernel on a non-compact Riemannian manifold. Proc. Sympos. Pure Math., 57 (1995), 239–263.
  • Helgason, S., Groups and Geometric Analysis. Pure Appl. Math., 113. Academic Press, Orlando, FL, 1984.
  • Järvi, P. & Vuorinen, M., Uniformly perfect sets and quasiregular mappings. J. London Math. Soc., 54 (1996), 515–529.
  • Jørgensen, T. & Marden, A., Algebraic and geometric convergence of Kleinian groups. Math. Scand., 66 (1990), 47–72.
  • Kra, I. & Maskit, B., The deformation space of a Kleinian group. Amer. J. Math., 103 (1981), 1065–1102.
  • Larman, D. H., On the Besicovitch dimension of the residual set of arbitrary packed disks in the plane. J. London Math. Soc., 42 (1967), 292–302.
  • Maskit, B., On boundaries of Teichmüller spaces and on Kleinian groups, II. Ann. of Math., 91 (1970), 607–639.
  • —, Kleinian Groups. Grundlehren Math. Wiss. 287. Springer-Verlag, Berlin-New York, 1988.
  • McMullen, C., Cusps are dense. Ann. of Math., 133 (1991), 217–247.
  • McShane, G., Parker, J. R. & Redfern, I., Drawing limit sets of Kleinian groups using finite state automata. Experiment. Math., 3 (1994), 153–170.
  • Mostow, G., Strong Rigidity of Locally Symmetric Spaces. Ann. of Math. Stud., 78. Princeton Univ. Press, Princeton, NJ, 1973.
  • Nicholls, P. J., The limit set of a discrete group of hyperbolic motions, in Holomorphic Functions and Moduli, Vol. II (Berkeley, CA, 1986), pp. 141–164. Math. Sci. Res. Inst. Publ., 11. Springer-Verlag, New York-Berlin, 1988.
  • The Ergodic Theory of Discrete Groups. London Math. Soc. Lecture Note Ser., 143. Cambridge Univ. Press, Cambridge, 1989.
  • Parker, J. R., Kleinian circle packings. Topology, 34 (1995), 489–496.
  • Patterson, S. J., The exponent of convergence of Poincaré series. Monatsh. Math., 82 (1976), 297–315.
  • — Lectures on measures on limit sets of Kleinian groups, in Analytical and Geometric Aspects of Hyperbolic Space, pp. 281–323. London Math. Soc. Lecture Note Ser., 111. Cambridge Univ. Press, Cambridge-New York, 1987.
  • Pommerenke, Ch., On uniformly perfect sets and Fuchsian groups. Analysis, 4 (1984), 299–321.
  • Stratmann, B., The Hausdorff dimension of bounded geodesics on geometrically finite manifolds. Ergodic Theory Dynamical Systems, 17 (1997), 227–246.
  • Stratmann, B. & Urbanski, M., The box counting dimension for geometrically finite Kleinian groups. Fund. Math., 149 (1996), 83–93.
  • Stratmann, B. & Velani, S. L., The Patterson measure for geometrically finite groups with parabolic elements, new and old. Proc. London Math. Soc., 71 (1995), 197–220.
  • Sullivan, D., The density at infinity of a discrete groups of hyperbolic motions. Inst. Hautes Études Sci. Publ. Math., 50 (1979), 172–202.
  • —, Growth of positive harmonic functions and Kleinian group limit sets of planar measure 0 and Hausdorff dimension 2, in Geometry Symposium (Utrecht, 1980), pp. 127–144. Lecture Notes in Math., 894. Springer-Verlag, Berlin-New York, 1981.
  • — Discrete conformal groups and measurable dynamics. Bull. Amer. Math. Soc., 6 (1982), 57–73.
  • —, Entropy, Hausdorff measures new and old and limit sets of geometrically finite Kleinian groups. Acta Math., 153 (1984), 259–277.
  • — Related aspects of positivity in Riemannian geometry. J. Differential Geom., 25 (1987), 327–351.
  • Taylor, E., On the volume of convex cores under algebraic and geometric convergence. PhD Thesis, SUNY at Stony Brook, 1994.
  • Tukia, P., The Hausdorff dimension of the limit set of a geometrically finite Kleinian group. Acta Math., 152 (1984), 127–140.
  • — On the dimension of limit sets of geometrically finite Möbius groups. Ann. Acad. Sci. Fenn. Ser. A I Math., 19 (1994), 11–24.
  • — The Poincaré series and the conformal measure of conical and Myrberg limit points. J. Analyse Math., 62 (1994), 241–259.