A noncompact chaotic billiard on a two-dimensional space of constant negative curvature, the infinite equilateral triangle describing anisotropy oscillations in the very early universe, is studied quantum mechanically. A Weyl formula with a logarithmic correction term is derived for the smoothed number of states function. For one symmetry class of the eigenfunctions the level spacing distribution, the spectral rigidity Δ 3, and the Σ 2 statistics are determined numerically using the finite matrix approximation. Systematic deviations are found both from the Gaussian orthogonal ensemble (GOE) and the Poissonian ensemble. However, good agreement with the GOE is found if the fundamental triangle is deformed in such a way that it no longer tiles the space.