Volume 18, issue 3 (2018)

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Macfarlane hyperbolic $3$–manifolds

Joseph A Quinn

Algebraic & Geometric Topology 18 (2018) 1603–1632

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.

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Macfarlane space, quaternion hyperboloid, hyperbolic quaternions
Mathematical Subject Classification 2010
Primary: 11R52, 57M27, 57M99
Received: 4 March 2017
Revised: 15 January 2018
Accepted: 12 February 2018
Published: 3 April 2018
Joseph A Quinn
Instituto de Matemáticas, Unidad Cuernavaca