Duke Mathematical Journal

Hyperbolic convex cores and simplicial volume

Peter A. Storm

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This article investigates the relationship between the topology of hyperbolizable 3-manifolds M with incompressible boundary and the volume of hyperbolic convex cores homotopy equivalent to M. Specifically, it proves a conjecture of Bonahon stating that the volume of a convex core is at least half the simplicial volume of the doubled manifold DM, and this inequality is sharp. This article proves that the inequality is, in fact, sharp in every pleating variety of AH(M)

Article information

Duke Math. J., Volume 140, Number 2 (2007), 281-319.

First available in Project Euclid: 18 October 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Secondary: 57N10: Topology of general 3-manifolds [See also 57Mxx]


Storm, Peter A. Hyperbolic convex cores and simplicial volume. Duke Math. J. 140 (2007), no. 2, 281--319. doi:10.1215/S0012-7094-07-14023-7. https://projecteuclid.org/euclid.dmj/1192715421

Export citation


  • I. Agol, Topology of hyperbolic $3$-manifolds, Ph.D. dissertation, University of California, San Diego, La Jolla, Calif., 1998.
  • J. W. Anderson and R. D. Canary, Cores of hyperbolic $3$-manifolds and limits of Kleinian groups, II, J. London Math. Soc. (2) 61 (2000), 489--505.
  • R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry, Universitext, Springer, Berlin, 1992.
  • L. BessièRes, Sur le volume minimal des variétés ouvertes, Ann. Inst. Fourier (Grenoble) 50 (2000), 965--980.
  • G. Besson, G. Courtois, and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal. 5 (1995), 731--799.
  • F. Bonahon, A Schläfli-type formula for convex cores of hyperbolic $3$-manifolds, J. Differential Geom. 50 (1998), 25--58.
  • F. Bonahon and J.-P. Otal, Laminations measurées de plissage des variétés hyperboliques de dimension $3$, Ann. of Math. (2) 160 (2004), 1013--1055.
  • B. H. Bowditch, Some results on the geometry of convex hulls in manifolds of pinched negative curvature, Comment. Math. Helv. 69 (1994), 49--81.
  • M. Bridgeman and R. D. Canary, From the boundary of the convex core to the conformal boundary, Geom. Dedicata 96 (2003), 211--240.
  • D. Burago, Y. Burago, and S. Ivanov, A Course in Metric Geometry, Grad. Stud. Math. 33, Amer. Math. Soc., Providence, 2001.
  • Yu. Burago, M. Gromov, and G. Perel'Man, A. D. Aleksandrov spaces with curvatures bounded below (in Russian), Uspekhi Mat. Nauk 47, no. 2 (1992), 3--51.; English translation in Russian Math. Surveys 47, no. 2 (1992), 1--58.
  • R. D. Canary, ``Covering theorems for hyperbolic $3$-manifolds'' in Low-Dimensional Topology (Knoxville, Tenn., 1992), Conf. Proc. Lecture Notes Geom. Topology 3, Int. Press, Cambridge, Mass., 1994, 21--30.
  • R. D. Canary, Y. N. Minsky, and E. C. Taylor, Spectral theory, Hausdorff dimension and the topology of hyperbolic $3$-manifolds, J. Geom. Anal. 9 (1999), 17--40.
  • 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, U.K., 1984), London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press, Cambridge, 1987, 113--253.
  • C. D. Hodgson and S. P. Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery, J. Differential Geom. 48 (1998), 1--59.
  • W. H. Jaco, Lectures on Three-Manifold Topology, CBMS Regional Conf. Ser. Math. 43, Amer. Math. Soc., Providence, 1980.
  • W. H. Jaco and P. B. Shalen, Seifert Fibered Spaces in $3$-Manifolds, Mem. Amer. Math. Soc. 21 (1979), no. 220.
  • K. Johannson, Homotopy Equivalences of $3$-manifolds with Boundaries, Lecture Notes in Math. 761, Springer, Berlin, 1979.
  • T. JøRgensen and A. Marden, Algebraic and geometric convergence of Kleinian groups, Math. Scand. 66 (1990), 47--72.
  • S. P. Kerckhoff and W. P. Thurston, Noncontinuity of the action of the modular group at Bers' boundary of Teichmüller space, Invent. Math. 100 (1990), 25--47.
  • C. Lecuire, Plissage des variétés hyperboliques de dimension $3$, Invent. Math. 164 (2006), 85--141.
  • B. Leeb, $3$-manifolds with(out) metrics of nonpositive curvature, Invent. Math. 122 (1995), 277--289.
  • C. T. Mcmullen, Renormalization and $3$-Manifolds Which Fiber Over the Circle, Ann. of Math. Stud. 142, Princeton Univ. Press, Princeton, 1996.
  • —, Hausdorff dimension and conformal dynamics, I: Strong convergence of Kleinian groups, J. Differential Geom. 51 (1999), 471--515.
  • J. W. Morgan, ``On Thurston's uniformization theorem for three-dimensional manifolds'' in The Smith Conjecture (New York, 1979), Pure Appl. Math. 112, Academic Press, Orlando, Fla., 1984, 37--125.
  • G. Perelman, Alexandrov's spaces with curvatures bounded from below, II, preprint, 1991.
  • C. Series, Limits of quasi-Fuchsian groups with small bending, Duke Math. J. 128 (2005), 285--329.
  • T. Soma, The Gromov invariant of links, Invent. Math. 64 (1981), 445--454.
  • J. Souto, Geometric structures on $3$-manifolds and their deformations, Ph.D. dissertation, Rheinischen Friedrich-Wilhelms-Universität Bonn, Bonn, Germany, 2001.
  • P. A. Storm, Minimal volume Alexandrov spaces, J. Differential Geom. 61 (2002), 195--225.
  • —, The barycenter method on singular spaces, Comment. Math. Helv. 82 (2007), 133--173.
  • E. C. Taylor, Geometric finiteness and the convergence of Kleinian groups, Comm. Anal. Geom. 5 (1997), 497--533.
  • W. P. Thurston, ``Hyperbolic geometry and $3$-manifolds'' in Low-Dimensional Topology (Bangor, Wales, 1979), London Math. Soc. Lecture Note Ser. 48, Cambridge Univ. Press, Cambridge, 1982, 9--25.
  • —, Hyperbolic structures on 3-manifolds, I: Deformation of acylindrical manifolds, Ann. of Math. (2) 124 (1986), 203--246.
  • —, The geometry and topology of three-manifolds, lecture notes, Math. Dept., Princeton Univ., Princeton, 1980.
  • —, Hyperbolic structures on $3$-manifolds, III: Deformations of, $3$-manifolds with incompressible boundary, preprint,\arxivmath/9801058v1[math.GT]