\(H^3\) as a harmonic branched covering of \(E^3\)
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.
scientific article; zbMATH DE number 6835198
scientific article