Periodic Polyhedra in Spaces of Constant Curvature

Abstract

We show the existence of families of periodic polyhedra in spaces of constant curvature whose fundamental domains can be obtained by attaching prisms and antiprisms to Archimedean solids. These polyhedra have constant discrete curvature and are weakly regular in the sense that all faces are congruent regular polygons and all vertex figures are congruent as well. Some of our examples have stronger conformal or metric regularity. The polyhedra are invariant under either a group generated by reflections at the faces of a Platonic solid, or a group generated by transformations that are reflections at the faces of a Platonic solid, followed by a rotation about an axis perpendicular to the respective face. In particular, suitable quotients will be compact polyhedral surfaces in (possibly non-compact) spaceforms.