# Macfarlane Hyperbolic 3-Manfiolds

## TL;DR

This paper introduces Macfarlane hyperbolic 3-manifolds, characterizes their arithmetic properties, and develops a new method for computing their Dirichlet domains, with applications to hyperbolic surfaces.

## Contribution

It defines Macfarlane manifolds, characterizes their arithmetic nature, and presents a novel computational approach for their Dirichlet domains.

## Key findings

- Infinitely many commensurability classes of Macfarlane manifolds are identified.
- A new method for computing Dirichlet domains of these manifolds is introduced.
- Hyperbolic surfaces related to Macfarlane manifolds are characterized and studied.

## 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 also give similar results for a class of hyperbolic surfaces and explore their occurrence as subsurfaces of Macfarlane manifolds.