title: Hyperbolic crystallography of two-periodic surfaces and associated structures.
abstract: 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 {\sf {2223}} and a {\sf {22222}} surface decoration.
authors.affiliation: Department of Applied Mathematics, Research School of Physics and Engineering, Australian National University, Canberra ACT 2601, Australia.
authors.forename: Martin Cramer
authors.initials: MC
authors.lastname: Pedersen
authors.affiliation: Department of Applied Mathematics, Research School of Physics and Engineering, Australian National University, Canberra ACT 2601, Australia.
authors.forename: Stephen T
authors.initials: ST
authors.lastname: Hyde
keywords: constant mean curvature surfaces; hyperbolic crystallography; hyperbolic geometry