A class of hyperbolic space groups (Q1179572)

In the threedimensional hyperbolic space there is no classification of the discrete motion groups. Thus the author asks for motion groups in this space possessing a combinatorically given polyhedron \(P\) as a fundamental domain. In this paper he considers polyhedra being a regular tetrahedron. He studies the metrical realizations of the tetrahedron with which the hyperbolic 3-space can be filled out. With this note he continues several papers of himself. --- An identification of two facets of \(P\) generates a discrete motion group. There exist exactly 7 sorts of identifications where all the edges of the identified tetrahedron are equivalent. A realization in the hyperbolic 3- space is only possible if the vertices of \(P\) are ideal. Therefore the author gets 7 combinatorically different identifications with the vertices on the absolute. For vertices outside of the absolute the tetrahedron must be truncated by the polar hyperplanes of the vertices. This is necessary to get a compact fundamental domain. Thus the author gets further exactly 12 different space groups having truncated tetrahedra as fundamental domains. All these 19 hyperbolic space groups and their fundamental domains are listed in a table, and the corresponding figures are drawn.