Hyperbolic crystallography of two-periodic surfaces and associated structures

This paper describes the families of the simplest, two-periodic constant mean curvature surfaces, the genus-two HCB and SQL surfaces, and their isometries. All the discrete groups that contain the translations of the genus-two surfaces embedded in Euclidean three-space modulo the translation lattice are derived and enumerated. Using this information, the subgroup lattice graphs are constructed, which contain all of the group-subgroup relations of the aforementioned quotient groups. The resulting groups represent the two-dimensional representations of subperiodic layer groups with square and hexagonal supergroups, allowing exhaustive enumeration of tilings and associated patterns on these surfaces. Two examples are given: a two-periodic [3,7]-tiling with hyperbolic orbifold symbol and a surface decoration.The intrinsic, hyperbolic crystallography of the two-periodic, genus-two HCB and SQL surfaces is presented. All discrete groups containing the translations of the Euclidean embeddings of these surfaces are derived and examples of applications are given.