Tameness on the boundary and Ahlfors' measure conjecture
type: Item
instance of: scholarly article
title: Tameness on the boundary and Ahlfors' measure conjecture
zbMATH Open document ID: 1060.30054
author: Jeffrey F. Brock
author: Richard Evans
author: Juan Souto
author: Kenneth W. Bromberg
published in: Publications Mathématiques
publication date: 2004-10-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0211022
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104194
review text: A complete hyperbolic 3-manifold is said to be tame if it is homeomorphic to the interior of a compact 3-manifold. Marden's tameness conjecture is that a complete hyperbolic 3-manifold with finitely generated fundamental group is tame. In this paper the authors show that a complete hyperbolic 3-manifold \(N\) which is an algebraic limit of geometrically finite hyperbolic 3-manifolds is tame if \(N\) has non-empty conformal boundary. This result reduces the Ahlfors' measure conjecture to the following density conjecture: a complete hyperbolic 3-manifold with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3-manifolds. The key theorem is that an algebraic limit of geometrically finite hyperbolic 3-manifolds is a limit of a type-preserving sequence of geometrically finite hyperbolic 3-manifolds. The authors also show the tameness with respect to a compression body and a strong limit of geometrically finite manifolds.
reviewed by: Gou Nakamura
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 2111679
zbMATH Keywords: Marden's tameness conjecture
zbMATH Keywords: Ahlfors measure conjecture
zbMATH Keywords: hyperbolic 3-manifold
MaRDI profile type: Publication
OpenAlex ID: W2049310833
Wikidata QID: Q122954856
arXiv ID: math/0211022
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DOI: 10.1007/S10240-003-0018-Y
description: scientific article