Hyperbolic space groups and edge conditions for their domains

TL;DR

This paper investigates conditions under which simplicial fundamental domains of hyperbolic space groups, specifically those in Family F12, can be realized with vertices outside the absolute, using edge conditions.

Contribution

It introduces new edge conditions for simplicial fundamental domains to determine hyperbolicity with vertices outside the absolute in hyperbolic space groups.

Findings

Edge conditions restrict realizability of fundamental domains in hyperbolic space.

Simplicial fundamental domains in Family F12 can be characterized by these edge conditions.

Results help identify when such domains are hyperbolic with vertices out of the absolute.

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. Moln\'ar et all in 2006, are considered.