Quaternionic Kleinian modular groups and arithmetic hyperbolic orbifolds over the quaternions

TL;DR

This paper explores quaternionic Kleinian groups derived from Lipschitz and Hurwitz integers, constructing discrete subgroups of PSL(2, H) to study their properties and associated hyperbolic orbifolds.

Contribution

It introduces new Kleinian groups over quaternions using Lipschitz and Hurwitz integers, expanding the understanding of quaternionic hyperbolic geometry.

Findings

Construction of new quaternionic Kleinian groups

Analysis of their arithmetic properties

Connections to hyperbolic orbifolds

Abstract

Using the rings of Lipschitz and Hurwitz integers $\mathbb{H}(\mathbb{Z})$ and $\mathbb{H}ur(\mathbb{Z})$ in the quaternion division algebra $\mathbb{H}$, we define several Kleinian discrete subgroups of $PSL(2,\mathbb{H})$