Hyperbolic space groups and edge conditions for their domains

Abstract

Looking to the fundamental domains of space groups we can investigate in which space they can be realized. If this space is hyperbolic, then the corresponding space group is also hyperbolic. In addition to the usual methods for investigating space of realization, the symmetries of the fundamental polyhedron can give new restricted conditions, here called edge conditions. The aim of the research is to find out in which cases simplicial fundamental domains are hyperbolic with vertices out of the absolute. For this reason, edge conditions for simplicial fundamental domains belonging to Family F12 by the notation of E. Molnár et all in 2006, are considered.