Cusp shapes of Hilbert–Blumenthal surfaces


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.

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This research paper has been made possible thanks to the financial support generously given by the FORDECyT-CONACyT (Mexico) Grant #265667, Universidad Nacional Autónoma de México. The second author was financed by Grant IN106817, PAPIIT, DGAPA, Universidad Nacional Autónoma de México. The authors also express their gratitude to Ian Agol, Kathleen Byrne, Jesse Ira Deutsch, Paul Garrett, Ben McReynolds, Jorge Millan and Walter Neumann for helpful suggestions and discussion; to the reviewer for their detailed comments and corrections; and to Dennis Ryan and Simon Woods for help with creating the computer generated images.

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Correspondence to Alberto Verjovsky.

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Quinn, J., Verjovsky, A. Cusp shapes of Hilbert–Blumenthal surfaces. Geom Dedicata 206, 27–42 (2020).

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  • Hilbert modular surfaces
  • Topological manifolds
  • Geometric structures on manifolds
  • Algebraic numbers
  • Rings of algebraic integers
  • Real and complex geometry
  • Geometric constructions

Mathematics Subject Classification

  • 14G35