On Brillouin zones

Brillouin zones were introduced by Brillouin [Br] in the thirties to describe quantum mechanical properties of crystals, that is, in a lattice in ℝⁿ. They play an important role in solid-state physics. It was shown by Bieberbach [Bi] that Brillouin zones tile the underlying space and that each zone has the same area. We generalize the notion of Brillouin zones to apply to an arbitrary discrete set in a proper metric space, and show that analogs of Bieberbach's results hold in this context. We then use these ideas to discuss focusing of geodesics in spaces of constant curvature. In the particular case of the Riemann surfaces ℍ²/ Γ(k) (k = 2, 3, or 5), we explicitly count the number of geodesics of length t that connect the point i to itself.