Tameness on the boundary and Ahlfors' measure conjecture

Jeffrey Brock; Kenneth Bromberg; Richard Evans; Juan Souto

Publications Mathématiques de l'IHÉS (2003)

  • Volume: 98, page 145-166
  • ISSN: 0073-8301

Abstract

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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.

How to cite

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Brock, Jeffrey, et al. "Tameness on the boundary and Ahlfors' measure conjecture." Publications Mathématiques de l'IHÉS 98 (2003): 145-166. <http://eudml.org/doc/104194>.

@article{Brock2003,
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.},
author = {Brock, Jeffrey, Bromberg, Kenneth, Evans, Richard, Souto, Juan},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {Marden's tameness conjecture; Ahlfors measure conjecture; hyperbolic 3-manifold},
language = {eng},
pages = {145-166},
publisher = {Springer},
title = {Tameness on the boundary and Ahlfors' measure conjecture},
url = {http://eudml.org/doc/104194},
volume = {98},
year = {2003},
}

TY - JOUR
AU - Brock, Jeffrey
AU - Bromberg, Kenneth
AU - Evans, Richard
AU - Souto, Juan
TI - Tameness on the boundary and Ahlfors' measure conjecture
JO - Publications Mathématiques de l'IHÉS
PY - 2003
PB - Springer
VL - 98
SP - 145
EP - 166
AB - 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.
LA - eng
KW - Marden's tameness conjecture; Ahlfors measure conjecture; hyperbolic 3-manifold
UR - http://eudml.org/doc/104194
ER -

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