Tameness on the boundary and Ahlfors’ measure conjecture Article Received: 07 October 2002 DOI:
Cite this article as: Brock, J., Bromberg, K., Evans, R. et al. Publ. Math. (2003) 98: 145. doi:10.1007/s10240-003-0018-y Abstract
Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the interior of a compact 3-manifold, or
tame, if one of the following conditions holds:
1. N has non-empty conformal boundary,
2. N is not homotopy equivalent to a compression body, or
3. N is a strong limit of geometrically finite manifolds.
The first case proves Ahlfors’ measure conjecture for Kleinian groups in the closure of the geometrically finite locus: given any algebraic limit Γ of geometrically finite Kleinian groups, the limit set of Γ is either of Lebesgue measure zero or all of Ĉ. Thus, Ahlfors’ conjecture is reduced to the density conjecture of Bers, Sullivan, and Thurston.
W. Abikoff, Degenerating families of Riemann surfaces,
L. Ahlfors, Finitely generated Kleinian groups,
Am. J. Math.,
L. Ahlfors, Fundamental polyhedrons and limit point sets of Kleinian groups,
Proc. Natl. Acad. Sci. USA,
J. Anderson and R. Canary, Cores of hyperbolic 3-manifolds and limits of Kleinian groups,
Am. J. Math.,
J. Anderson and R. Canary, Cores of hyperbolic 3-manifolds and limits of Kleinian groups II,
J. Lond. Math. Soc.,
R. Benedetti and C. Petronio,
Lectures on Hyperbolic Geometry
, Springer-Verlag, 1992.
F. Bonahon, Cobordism of automorphisms of surfaces,
Ann. Sci. Éc. Norm. Supér.,
F. Bonahon, Bouts des variétés hyperboliques de dimension 3,
F. Bonahon and J. P. Otal, Variétés hyperboliques à géodésiques arbitrairement courtes,
Bull. Lond. Math. Soc.,
J. Brock, Iteration of mapping classes and limits of hyperbolic 3-manifolds,
J. Brock and K. Bromberg, Cone Manifolds and the Density Conjecture,
To appear in the proceedings of the Warwick conference ‘Kleinian groups and hyperbolic 3-manifolds,’ arXiv:mathGT/0210484
J. Brock and K. Bromberg, On the density of geometrically finite Kleinian groups,
Accepted to Acta Math.,
K. Bromberg, Hyperbolic Dehn surgery on geometrically infinite 3-manifolds,
K. Bromberg, Rigidity of geometrically finite hyperbolic cone-manifolds,
To appear, Geom. Dedicata,
K. Bromberg, Hyperbolic cone manifolds, short geodesics, and Schwarzian derivatives,
K. Bromberg, Projective structures with degenerate holonomy and the Bers density conjecture,
R. Brooks and J. P. Matelski, Collars for Kleinian Groups,
Duke Math. J.,
R. D. Canary, Ends of hyperbolic 3-manifolds,
J. Am. Math. Soc.,
R. D. Canary, Geometrically tame hyperbolic 3-manifolds, In
Differential geometry: Riemannian geometry (Los Angeles, CA, 1990)
, volume 54 of
Proc. Symp. Pure Math.
, pp. 99–109. Am. Math. Soc., 1993.
R. D. Canary, D. B. A. Epstein and P. Green, Notes on notes of Thurston. In
Analytical and Geometric Aspects of Hyperbolic Space
, pp. 3–92. Cambridge University Press, 1987.
R. D. Canary and Y. N. Minsky, On limits of tame hyperbolic 3-manifolds,
J. Differ. Geom.,
C. J. Earle and A. Marden, Geometric complex coordinates for Teichmüller space,
R. Evans, Deformation spaces of hyperbolic 3-manifolds: strong convergence and tameness,
Ph.D. Thesis, Unversity of Michigan (2000)
R. Evans. Tameness persists in weakly type-preserving strong limits,
C. Hodgson and S. Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery,
J. Differ. Geom.,
C. Hodgson and S. Kerckhoff, Universal bounds for hyperbolic Dehn surgery,
C. Hodgson and S. Kerckhoff, The shape of hyperbolic Dehn surgery space,
In preparation (2002)
S. Kerckhoff and W. Thurston, Non-continuity of the action of the modular group at Bers’ boundary of Teichmüller space,
R. Kulkarni and P. Shalen, On Ahlfors’ finiteness theorem,
A. Marden, The geometry of finitely generated kleinian groups,
A. Marden, Geometrically finite Kleinian groups and their deformation spaces, In
Discrete groups and automorphic functions
, Academic Press (1977), pp. 259–293. Academic Press, 1977.
B. Maskit, On boundaries of Teichmüller spaces and on kleinian groups: II,
, Springer-Verlag, 1988.
D. McCullough, Compact submanifolds of 3-manifolds with boundary,
Quart. J. Math. Oxf.,
C. McMullen, The classification of conformal dynamical systems, In
Current Developments in Mathematics, 1995
, pp. 323–360. International Press, 1995.
Renormalization and 3-Manifolds Which Fiber Over the Circle
, Annals of Math Studies 142, Princeton University Press, 1996.
Strong rigidity of locally symmetric spaces
, Annals of Math. Studies 78, Princeton University Press, 1972.
K. Ohshika, Kleinian groups which are limits of geometrically finite groups,
P. Scott, Compact submanifolds of 3-manifolds,
J. Lond. Math. Soc.,
D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, In
Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference
. Annals of Math. Studies 97, Princeton, 1981.
D. Sullivan, Quasiconformal homeomorphisms and dynamics II: Structural stability implies hyperbolicity for Kleinian groups,
W. P. Thurston,
Geometry and Topology of Three-Manifolds
, Princeton Lecture Notes, 1979.
W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry,
Bull. Am. Math. Soc.,
W. P. Thurston, Hyperbolic structures on 3-manifolds I: Deformations of acylindrical manifolds,
W. P. Thurston, Hyperbolic structures on 3-manifolds II: Surface groups and 3-manifolds which fiber over the circle,
Google Scholar Copyright information
© Institut des Hautes Études Scientifiques and Springer-Verlag 2003