\(H^3\) as a harmonic branched covering of \(E^3\)
type: Item
instance of: scholarly article
title: \(H^3\) as a harmonic branched covering of \(E^3\)
zbMATH Open document ID: 1393.57006
author: Enrique Ramírez-Losada
author: Hamish Short
author: Gregory W. Brumfiel
author: Hugh Michael Hilden
author: José María Montesinos-Amilibia
author: Débora María Tejada
author: Margarita María Toro
author: María Teresa Lozano Imízcoz
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2018-02-06T00:00:00Z
review text: The authors construct a \textit{harmonic} branched covering map \(p\): \(H^3\) \( \rightarrow \) \(E^3\) between hyperbolic and Euclidean space. The map is defined by using the orbifold group \(U\) of the Borromean rings, a Euclidean crystallographic group \(\hat{U}\), a surjective homomorphism \( \phi : U \rightarrow \hat{U}\) and triangulations of the dodecahedron and the cube. The groups \(U\), \( \hat{U}\) are given by their respective presentations, which arise from rotations of the dodecahedron and the cube, respectively. Some subgroups and supergroups of \(U\), \( \hat{U}\) are defined, which help to prove the fact that the map \(p\) can be chosen harmonic (i.e. a map which minimizes energy in the sense of the calculus of variations). Super regular tessellations of Euclidean and hyperbolic planes are discussed at length. The quotient spaces \(H^3 / U\) and \(E^3 / \hat{U}\) turn out to be topological 3-spheres.
reviewed by: Keerti Vardhan
Mathematics Subject Classification ID: 57M12
zbMATH DE Number: 6835198
zbMATH Keywords: branched coverings
zbMATH Keywords: harmonic functions
zbMATH Keywords: hyperbolic space
zbMATH Keywords: super regular tessellation
MaRDI profile type: Publication
full work available at URL: s40590-016-0114-x
OpenAlex ID: W2337992103
cites work: Q4440462
cites work: Three manifolds as geometric branched coverings of the three sphere
cites work: On finite index subgroups of a universal group
cites work: Harmonic Mappings of Riemannian Manifolds
cites work: Q5423215
cites work: On universal groups and three-manifolds
cites work: Q4337982
DOI: 10.1007/S40590-016-0114-X
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description: scientific article; zbMATH DE number 6835198