Hyperbolic convex cores and simplicial volume
Peter A. Storm
Source: Duke Math. J. Volume 140, Number 2 (2007), 281-319.
Abstract
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$(M)$
Primary Subjects: 53C25
Secondary Subjects: 57N10
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.dmj/1192715421
Digital Object Identifier: doi:10.1215/S0012-7094-07-14023-7
Mathematical Reviews number (MathSciNet):
MR2359821
References
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.
Mathematical Reviews (MathSciNet):
MR1760675
Digital Object Identifier: doi:10.1112/S0024610799008595
R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry, Universitext, Springer, Berlin, 1992.
Mathematical Reviews (MathSciNet):
MR1219310
L. BessièRes, Sur le volume minimal des variétés ouvertes, Ann. Inst. Fourier (Grenoble) 50 (2000), 965--980.
Mathematical Reviews (MathSciNet):
MR1779901
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.
Mathematical Reviews (MathSciNet):
MR1354289
Digital Object Identifier: doi:10.1007/BF01897050
F. Bonahon, A Schläfli-type formula for convex cores of hyperbolic $3$-manifolds, J. Differential Geom. 50 (1998), 25--58.
Mathematical Reviews (MathSciNet):
MR1678473
Project Euclid: euclid.jdg/1214510045
F. Bonahon and J.-P. Otal, Laminations measurées de plissage des variétés hyperboliques de dimension $3$, Ann. of Math. (2) 160 (2004), 1013--1055.
Mathematical Reviews (MathSciNet):
MR2144972
Digital Object Identifier: doi:10.4007/annals.2004.160.1013
B. H. Bowditch, Some results on the geometry of convex hulls in manifolds of pinched negative curvature, Comment. Math. Helv. 69 (1994), 49--81.
Mathematical Reviews (MathSciNet):
MR1259606
Digital Object Identifier: doi:10.1007/BF02564474
M. Bridgeman and R. D. Canary, From the boundary of the convex core to the conformal boundary, Geom. Dedicata 96 (2003), 211--240.
Mathematical Reviews (MathSciNet):
MR1956842
Digital Object Identifier: doi:10.1023/A:1022102007948
D. Burago, Y. Burago, and S. Ivanov, A Course in Metric Geometry, Grad. Stud. Math. 33, Amer. Math. Soc., Providence, 2001.
Mathematical Reviews (MathSciNet):
MR1835418
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.
Mathematical Reviews (MathSciNet):
MR1185284
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.
Mathematical Reviews (MathSciNet):
MR1316167
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.
Mathematical Reviews (MathSciNet):
MR1760718
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.
Mathematical Reviews (MathSciNet):
MR0903852
C. D. Hodgson and S. P. Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery, J. Differential Geom. 48 (1998), 1--59.
Mathematical Reviews (MathSciNet):
MR1622600
Project Euclid: euclid.jdg/1214460606
W. H. Jaco, Lectures on Three-Manifold Topology, CBMS Regional Conf. Ser. Math. 43, Amer. Math. Soc., Providence, 1980.
Mathematical Reviews (MathSciNet):
MR0565450
W. H. Jaco and P. B. Shalen, Seifert Fibered Spaces in $3$-Manifolds, Mem. Amer. Math. Soc. 21 (1979), no. 220.
Mathematical Reviews (MathSciNet):
MR0539411
K. Johannson, Homotopy Equivalences of $3$-manifolds with Boundaries, Lecture Notes in Math. 761, Springer, Berlin, 1979.
Mathematical Reviews (MathSciNet):
MR0551744
T. JøRgensen and A. Marden, Algebraic and geometric convergence of Kleinian groups, Math. Scand. 66 (1990), 47--72.
Mathematical Reviews (MathSciNet):
MR1060898
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.
Mathematical Reviews (MathSciNet):
MR1037141
Digital Object Identifier: doi:10.1007/BF01231179
C. Lecuire, Plissage des variétés hyperboliques de dimension $3$, Invent. Math. 164 (2006), 85--141.
Mathematical Reviews (MathSciNet):
MR2207784
Digital Object Identifier: doi:10.1007/s00222-005-0470-z
B. Leeb, $3$-manifolds with(out) metrics of nonpositive curvature, Invent. Math. 122 (1995), 277--289.
Mathematical Reviews (MathSciNet):
MR1358977
Digital Object Identifier: doi:10.1007/BF01231445
C. T. Mcmullen, Renormalization and $3$-Manifolds Which Fiber Over the Circle, Ann. of Math. Stud. 142, Princeton Univ. Press, Princeton, 1996.
Mathematical Reviews (MathSciNet):
MR1401347
—, Hausdorff dimension and conformal dynamics, I: Strong convergence of Kleinian groups, J. Differential Geom. 51 (1999), 471--515.
Mathematical Reviews (MathSciNet):
MR1726737
Project Euclid: euclid.jdg/1214425139
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.
Mathematical Reviews (MathSciNet):
MR0758464
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.
Mathematical Reviews (MathSciNet):
MR2140265
Digital Object Identifier: doi:10.1215/S0012-7094-04-12823-4
Project Euclid: euclid.dmj/1117728417
T. Soma, The Gromov invariant of links, Invent. Math. 64 (1981), 445--454.
Mathematical Reviews (MathSciNet):
MR0632984
Digital Object Identifier: doi:10.1007/BF01389276
J. Souto, Geometric structures on $3$-manifolds and their deformations, Ph.D. dissertation, Rheinischen Friedrich-Wilhelms-Universität Bonn, Bonn, Germany, 2001.
Mathematical Reviews (MathSciNet):
MR1934287
P. A. Storm, Minimal volume Alexandrov spaces, J. Differential Geom. 61 (2002), 195--225.
Mathematical Reviews (MathSciNet):
MR1972145
Project Euclid: euclid.jdg/1090351384
—, The barycenter method on singular spaces, Comment. Math. Helv. 82 (2007), 133--173.
Mathematical Reviews (MathSciNet):
MR2296060
E. C. Taylor, Geometric finiteness and the convergence of Kleinian groups, Comm. Anal. Geom. 5 (1997), 497--533.
Mathematical Reviews (MathSciNet):
MR1487726
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.
Mathematical Reviews (MathSciNet):
MR0662424
—, Hyperbolic structures on 3-manifolds, I: Deformation of acylindrical manifolds, Ann. of Math. (2) 124 (1986), 203--246.
Mathematical Reviews (MathSciNet):
MR0855294
Digital Object Identifier: doi:10.2307/1971277
—, 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]
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