Macfarlane hyperbolic 3–manifolds
Copyright policy )Macfarlane hyperbolic 3–manifolds
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 also give similar results for a class of hyperbolic surfaces and explore their occurrence as subsurfaces of Macfarlane manifolds.
22 pages, 4 figures, 1 table
arithmetic Kleinian groups, Structure of modular groups and generalizations; arithmetic groups, Geometric Topology (math.GT), Discrete subgroups of Lie groups, 57M99, Mathematics - Geometric Topology, quaternion hyperboloid, hyperbolic quaternions, 57M27, fundamental domain, General geometric structures on low-dimensional manifolds, FOS: Mathematics, quaternion algebra, Quaternion and other division algebras: arithmetic, zeta functions, Fuchsian groups and their generalizations (group-theoretic aspects), Macfarlane space, 11R52
arithmetic Kleinian groups, Structure of modular groups and generalizations; arithmetic groups, Geometric Topology (math.GT), Discrete subgroups of Lie groups, 57M99, Mathematics - Geometric Topology, quaternion hyperboloid, hyperbolic quaternions, 57M27, fundamental domain, General geometric structures on low-dimensional manifolds, FOS: Mathematics, quaternion algebra, Quaternion and other division algebras: arithmetic, zeta functions, Fuchsian groups and their generalizations (group-theoretic aspects), Macfarlane space, 11R52
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