Cusp shapes of Hilbert-Blumenthal surfaces

TL;DR

This paper constructs a new fundamental domain for the cusp stabilizer of Hilbert modular groups over real quadratic fields, revealing the cusp cross section's Sol 3-manifold structure and Anosov diffeomorphism.

Contribution

It introduces a novel fundamental domain as a union of Dirichlet domains over leaves in a foliation, explicitly describing the cusp structure of Hilbert-Blumenthal surfaces.

Findings

Explicit description of cusp cross sections as Sol 3-manifolds

Construction of fundamental domain as a union of Dirichlet domains

Visual illustrations and data for various examples

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.