Immersed surfaces in cubed manifolds
type: Item
instance of: scholarly article
title: Immersed surfaces in cubed manifolds
zbMATH Open document ID: 0935.57033
author: Iain R. Aitchison
author: S. Matsumotoi
author: J. Hyam Rubinstein
published in: The Asian Journal of Mathematics
publication date: 2000-05-03T00:00:00Z
review text: Let \(M\) be a compact, connected, oriented PL \(n\)-manifold which is the underlying space of a cube complex, and assume \(M\) has nonpositive curvature in the path-length metric determined by the cubing. Such an \(M\) contains a canonical immersed hypersurface \(S\) defined by the requirement that the intersection of \(S\) with every \(n\)-cube \(C\) of \(M\) be the union of the \(n\) codimension-1 cubes which contain the center of \(C\) and are parallel to a pair of \((n-1)\)-faces of \(C\), and this immersed hypersurface is \(\pi_1\)-injective. In the main theorem of this paper, the authors prove that if every \((n-2)\)-cube of the cubing of \(M\) is incident to an even number of \((n-1)\)-cubes, then there exists a regular \(n!\)-fold covering of \(M\) to which \(S\) lifts as an embedding. The motivation for this theorem comes from the work connected with the geometrization conjecture in 3-dimensional topology. Namely, \textit{I. R. Aitchison} and \textit{J. H. Rubinstein} [Lond. Math. Soc. Lect. Note Ser. 151, 127-161 (1990; Zbl 0735.57005)] have conjectured that every nonpositively curved cubed atoroidal 3-manifold \(M\) admits a complete hyperbolic structure, and they proved that this conjecture is true if \(M\) contains an immersed \(\pi_1\)-injective surface which lifts to an embedded surface in some finite cover of \(M\). The authors give a number of applications of their theorem, mainly to 3-dimensional topology, proving many 3-manifolds virtually Haken.
Mathematics Subject Classification ID: 57Q35
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 1310250
zbMATH Keywords: cubed manifold
zbMATH Keywords: nonpositive curvature
zbMATH Keywords: hyperbolic 3-manifold
zbMATH Keywords: atoroidal 3-manifold
zbMATH Keywords: virtually Haken manifold
zbMATH Keywords: immersed incompressible surface
zbMATH Keywords: Weber-Seifert dodecahedral space
zbMATH Keywords: balanced link
zbMATH Keywords: nice polyhedral decomposition
zbMATH Keywords: flying saucer decomposition
reviewed by: Joze Vrabec
MaRDI profile type: Publication
full work available at URL: ajm.1997.v1.n1.a4
OpenAlex ID: W2737026693
DOI: 10.4310/AJM.1997.V1.N1.A4
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description: scientific article; zbMATH DE number 1310250