Cusp shapes of Hilbert-Blumenthal surfaces
arXiv:1711.02418 · doi:10.1007/s10711-019-00474-w
Abstract
We introduce a new fundamental domain for the cusp stabilizer of a Hilbert modular group over a real quadratic field K=Q(sqrt n). This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of the biplane. The region is the Cartesian product of the positive reals with a 3-dimensional tower formed by deformations of lattices in the ring of integers of K, and makes explicit the cusp cross section's Sol 3-manifold structure and Anosov diffeomorphism. We include computer generated images and data illustrating various examples.
17 pages, 4 figures, 2 tables