On hyperbolic once-punctured-torus bundles
type: Item
instance of: scholarly article
title: On hyperbolic once-punctured-torus bundles
zbMATH Open document ID: 1015.57501
DOI: 10.1023/A:1020956906487
author: Warren Dicks
author: James W. Cannon
published in: Geometriae Dedicata
publication date: 2003-02-04T00:00:00Z
review text: Among the early examples of hyperbolic \(3\)-manifolds were certain bundles over the circle with surface fiber. The universal cover of the fiber is the hyperbolic disk \(H^2\), and the inclusion of the fiber lifts to an equivariant imbedding of \(H^2\) into \(H^3\). In an unpublished but highly influential 1989 preprint, the first author and Thurston showed that when the fiber is a closed surface, this inclusion extends to an equivariant surjective map from the circle at infinity to the sphere at infinity. When the fiber of the surface bundle \(M\) is not closed, each fiber in \(H^3\) has a spine which is a tree. \textit{C. T. McMullen} [Invent. Math. 146, No. 1, 35-91 (2001; Zbl 1061.37025)] showed that when the fiber of \(M\) is a once-punctured torus, the ends of this tree reach every point of the sphere at infinity. Deleting a single edge from this tree leaves two complementary subtrees. For the case when \(M\) is the figure-8 knot, \textit{R. C. Alperin, W. Dicks}, and \textit{J. Porti} [Topology Appl. 93, No.~3, 219-259 (1999; Zbl 0926.57008)] showed that the points in the sphere at infinity which are reached by the ends of one of these two subtrees form complementary closed disks. On the Jordan curve which is their intersection, the set of cusps is dense, in fact the curve is made up of two arcs with common endpoints such that the set of cusps in each arc is dense and forms a single orbit of a finitely generated semigroup of Möbius transformations. In the paper under review, these results are extended to all hyperbolic once-punctured torus bundles. The authors use a very algebraic approach to give a description of the set \({\mathcal E}\) of ends of the tree, and to develop a factorization \({\mathcal E}\to S^1\to S^2\), where \(S^2\) is a topological \(2\)-sphere. A result of \textit{B. H. Bowditch} [The Cannon-Thurston map for punctured-surface groups, preprint 2002] shows that this \(S^2\) is equivariantly homeomorphic to the \(2\)-sphere at infinity. The …
reviewed by: Darryl McCullough
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 1864098
zbMATH Keywords: Cannon-Thurston map
zbMATH Keywords: hyperbolic torus bundle
zbMATH Keywords: free group
zbMATH Keywords: fractal arc
zbMATH Keywords: cusp
zbMATH Keywords: dense
MaRDI profile type: Publication
full work available at URL: a:1020956906487
OpenAlex ID: W137235440
description: scientific article; zbMATH DE number 1864098