Flexible boundaries in deformations of hyperbolic 3-manifolds
type: Item
instance of: scholarly article
title: Flexible boundaries in deformations of hyperbolic 3-manifolds
zbMATH Open document ID: 0887.57016
author: Michihiko Fujii
author: Sadayoshi Kojima
published in: Osaka Journal of Mathematics
publication date: 1998-03-10T00:00:00Z
review text: For a complete finite-volume hyperbolic 3-manifold \(M\) with torus cusps Thurston considered deformations of the complete hyperbolic structure thus obtaining non-complete structures whose completions include almost all closed manifolds obtained by Dehn filling (surgery) on the cusps (the ``hyperbolic surgery theorem''). Now suppose that \(M\) has both cusps and compact totally geodesic boundary \(\partial M\). Then one may consider those deformations of the complete hyperbolic structure (that is complete on the double of \(M\) along \(\partial M)\) such that the boundary remains totally geodesic. By taking the induced hyperbolic structure on the boundary this defines a mapping \(B_M\) from the space of such deformations to the Teichmüller space of \(\partial M\). For dimension reasons, this mapping will not be injective in general. Neumann-Reid and the first author gave examples where it is constant. In contrast to this, in the present paper an infinite series of such manifolds \(M\) is constructed for which \(B_M\) is a local embedding at the complete structure. The manifolds are the \(n\)-fold cyclic branched coverings of hyperbolic orbifolds obtained from the complement of the Whitehead link by introducing a singular arc of index \(n\) whose endpoints are on the same component of the link. These orbifolds can be decomposed into 4 partially truncated ideal tetrahedra forming an octahedron. Then, using explicit parameters for the deformation space, it can be shown by a direct computation that the derivative of the map \(B_M\) has maximal rank.
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1117470
zbMATH Keywords: hyperbolic 3-manifold
zbMATH Keywords: deformation
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1117470