Sphere-packing and volume in hyperbolic 3-space
type: Item
instance of: scholarly article
title: Sphere-packing and volume in hyperbolic 3-space
zbMATH Open document ID: 0611.57010
DOI: 10.1007/BF02621915
author: Robert Meyerhoff
published in: Commentarii Mathematici Helvetici
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140052
review text: By a theorem of Jørgensen and Thurston [\textit{W. Thurston}, The geometry and topology of 3-manifolds, Princeton Univ., preprint (1978)], the set of volumes of complete hyperbolic 3-manifolds is well-ordered and of order type \(\omega^{\omega}\). In particular, there is a complete hyperbolic 3-manifold of minimum volume \(V_ 1\) among all complete hyperbolic 3-manifolds, and a cusped hyperbolic 3-manifold of minimum volume \(V_{\omega}\). Similar results hold for complete hyperbolic 3- orbifolds, implying the existence of a hyperbolic 3-orbifold of minimum volume \(V_ 1'\), and a cusped hyperbolic 3-orbifold of minimum volume \(V_ c'.\) The paper discusses inequalities for \(V_ 1\), \(V_{\omega}\), \(V_ 1'\) and \(V_ c'\); these are based on sphere packing arguments in hyperbolic 3-space \(H^ 3\). In particular it is shown that the orbifold \(Q_ 1=H^ 3/PGL_ 2({\mathcal O}_ 3)\) has minimum volume \(V_ c'=V/12\) among all orientable cusped hyperbolic 3-orbifolds; here, \({\mathcal O}_ 3\) is the ring of integers in \({\mathbb{Q}}(\sqrt{-3})\), and V is the volume of the ideal regular tetrahedron in \(H^ 3\).
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 51M10
zbMATH DE Number: 3988273
zbMATH Keywords: sphere packing in hyperbolic 3-space
zbMATH Keywords: hyperbolic 3-manifold of minimum volume
zbMATH Keywords: cusped hyperbolic 3-manifold of minimum volume
zbMATH Keywords: complete hyperbolic 3-orbifolds
zbMATH Keywords: hyperbolic 3-orbifold of minimum volume
zbMATH Keywords: cusped hyperbolic 3-orbifold of minimum volume
reviewed by: Egon Schulte
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3988273