Quaternion algebras and hyperbolic 3-manifolds

Joe Quinn (CUNY)

I reinterpret a classical idea of Macfarlane to obtain a complex quaternion model for hyperbolic 3-space and its group of orientation-preserving isometries, analogous to Hamilton's famous result on Euclidean rotations. I generalize this to quaternion models over number fields for the action of Kleinian groups on hyperbolic 3-space, using arithmetic invariants of their corresponding hyperbolic 3-manifolds. I develop new tools to study such manifolds, and focus on a new algorithm for computing their Dirichlet domains. This algorithm applies to all cusped arithmetic manifolds, many cusped non-arithmetic manifolds, and infinitely many commensurability classes of both arithmetic and non-arithmetic compact manifolds.