Cusp shapes of Hilbert–Blumenthal surfaces

Abstract

We introduce a new fundamental domain $\mathscr {R}_n$ for a cusp stabilizer of a Hilbert modular group $\Gamma$ over a real quadratic field $K=\mathbb {Q}(\sqrt{n})$. This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of $\mathcal {H}^2\times \mathcal {H}^2$. The region $\mathscr {R}_n$ is the product of $\mathbb {R}^+$ with a 3-dimensional tower $\mathcal {T}_n$ formed by deformations of lattices in the ring of integers $\mathbb {Z}_K$, 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.