Applications of group cohomology to the classification of quasicrystal symmetries

In 1962, Bienenstock and Ewald described the classification of crystalline space groups algebraically in the dual, or Fourier, space. After the discovery of quasicrystals in 1984, Mermin and collaborators recognized in this description the principle of macroscopic indistinguishability and developed techniques that have since been applied to quasicrystals, including also periodic and incommensurately modulated structures. This paper phrases these techniques in terms of group cohomology. A quasicrystal is defined, along with its space group, without requiring that it come from a quasicrystal in real (direct) space. A certain cohomology group classifies the space groups associated to a given point group and lattice, and the dual homology group gives all gauge invariants. This duality is exploited to prove several results that were previously known only in special cases, including the classification of space groups (plane groups) for lattices of arbitrary rank in two dimensions. Extinctions in x-ray diffraction patterns and degeneracy of electronic levels are interpreted as physical manifestations of non-zero homology classes. PACS numbers: 02.10.Hh, 61.50.Ah, 61.44.Br, 61.44.Fw Mathematics Subject Classification: 20H15, 20J06