We introduce a new fundamental domain Rn for a cusp stabilizer of a Hilbert modular group Γ over a real quadratic field K " Qp ? nq. This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of H 2ˆH2 . The region Rn is the product of R`with a 3-dimensional tower Tn formed by deformations of lattices in the ring of integers ZK , 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.
We identify and study a class of hyperbolic 3-manifolds (which we call Macfarlane manifolds) whose quaternion algebras admit a geometric interpretation analogous to Hamilton's classical model for Euclidean rotations. We characterize these manifolds arithmetically, and show that infinitely many commensurability classes of them arise in diverse topological and arithmetic settings. We then use this perspective to introduce a new method for computing their Dirichlet domains. We give similar results for a class of hyperbolic surfaces and explore their occurrence as subsurfaces of Macfarlane manifolds. Proposition 2.2. [30, §3.3] If K Ă C, then D a faithful matrix representation of´a ,b Kī nto M 2 pCq. Moreover, under any such representation, n and tr correspond to the matrix determinant and trace, respectively.
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