Pseudo-Anosov extensions and degree one maps between hyperbolic surface bundles
type: Item
instance of: scholarly article
title: Pseudo-Anosov extensions and degree one maps between hyperbolic surface bundles
zbMATH Open document ID: 1130.57023
author: Yi Ni
author: Michel Boileau
author: Shicheng Wang
published in: Mathematische Zeitschrift
publication date: 2007-06-07T00:00:00Z
full work available at URL: https://authors.library.caltech.edu/56805/
review text: For an orientable closed surface \(F\) of genus \(g\geq 1\) and an automorphism \(f:F\to F\), denote by \(M(F,f)\) the surface bundle with fiber \(F\) and monodromy \(f\). Work of Thurston implies that for \(g\geq 2\) the surface bundle \(M(F,f)\) is a hyperbolic \(3\)-manifold if and only if \(f\) is a pseudo-Anosov map (in the paper under review the definition of a pseudo-Anosov map is slightly different from the standard one: it refers to a map that is isotopic to a pseudo-Anosov map in the standard sense). Let now \(F'\) and \(F\) be two closed orientable surfaces of genus \(g'>g\geq 1\), respectively, and let \(f:F\to F\) be a pseudo-Anosov map. The main result states that \(f\) can be ``extended'' to a pseudo-Anosov mapping \(f':F'\to F'\) so that there is a fiber-preserving degree one map \(M(F',f')\to M(F,f)\) between the hyperbolic surface bundles. Furthermore, the ``extension'' can be chosen in such a way that the first Betti numbers of the two surface bundles agree.
reviewed by: Petra Bonfert-Taylor
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 5162377
zbMATH Keywords: pseudo-Anosov extension
zbMATH Keywords: hyperbolic surface bundles
MaRDI profile type: Publication
OpenAlex ID: W1994740582
cites work: Non-zero degree maps and surface bundles over \(S^ 1\)
cites work: Deformation of maps to branched coverings in dimension two
cites work: Hyperbolic 3-manifolds and surface automorphisms
cites work: Q4891081
cites work: Difféomorphismes pseudo-Anosov et automorphismes symplectiques de l'homologie
cites work: Three dimensional manifolds, Kleinian groups and hyperbolic geometry
cites work: Pseudo-Anosov maps and simple closed curves on surfaces
DOI: 10.1007/S00209-007-0113-8
description: scientific article; zbMATH DE number 5162377