Macfarlane hyperbolic 3–manifolds

Quinn, Joseph

doi:10.2140/agt.2018.18.1603

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Macfarlane hyperbolic $3$–manifolds

Joseph A Quinn

Algebraic & Geometric Topology 18 (2018) 1603–1632

DOI: 10.2140/agt.2018.18.1603

Abstract

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.

Keywords

Macfarlane space, quaternion hyperboloid, hyperbolic quaternions

Mathematical Subject Classification 2010

Primary: 11R52, 57M27, 57M99

References

Publication

Received: 4 March 2017

Revised: 15 January 2018

Accepted: 12 February 2018

Published: 3 April 2018

Authors

Joseph A Quinn
Instituto de Matemáticas, Unidad Cuernavaca UNAM Cuernavaca Mexico

Volume 18, issue 3 (2018)