  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.
