Macfarlane Hyperbolic 3-Manfiolds

摘要

We identify and study a class of hyperbolic 3-manifolds (which we callMacfarlane manifolds) whose quaternion algebras admit a geometricinterpretation analogous to Hamilton's classical model for Euclidean rotations.We characterize these manifolds arithmetically, and show that infinitely manycommensurability classes of them arise in diverse topological and arithmeticsettings. We then use this perspective to introduce a new method for computingtheir Dirichlet domains. We also give similar results for a class of hyperbolicsurfaces and explore their occurrence as subsurfaces of Macfarlane manifolds.