#### Volume 18, issue 3 (2018)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Macfarlane hyperbolic $3$–manifolds

### Joseph A Quinn

Algebraic & Geometric Topology 18 (2018) 1603–1632
##### Abstract

We identify and study a class of hyperbolic $3\phantom{\rule{-0.17em}{0ex}}$–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.

##### Keywords
Macfarlane space, quaternion hyperboloid, hyperbolic quaternions
##### Mathematical Subject Classification 2010
Primary: 11R52, 57M27, 57M99