On Units in Orders in 2-by-2 Matrices over Quaternion Algebras with Rational Center

TL;DR

This paper extends an algorithm to compute generators of finite index subgroups of arithmetic groups acting on hyperbolic spaces, specifically focusing on units in orders of 2x2 matrices over rational quaternion algebras.

Contribution

It generalizes an existing algorithm to higher-dimensional hyperbolic spaces using Clifford algebras, enabling the analysis of unit groups in orders over quaternion algebras.

Findings

Algorithm successfully computes generators for finite index subgroups.

Applicable to units in orders of 2x2 matrices over rational quaternion algebras.

Enhances understanding of arithmetic groups acting on hyperbolic spaces.

Abstract

We generalize an algorithm established in earlier work \cite{algebrapaper} to compute finitely many generators for a subgroup of finite index of an arithmetic group acting properly discontinuously on hyperbolic space of dimension $2$ and $3$, to hyperbolic space of higher dimensions using Clifford algebras. We hence get an algorithm which gives a finite set of generators of finite index subgroups of a discrete subgroup of Vahlen's group, i.e. a group of $2$-by-$2$ matrices with entries in the Clifford algebra satisfying certain conditions. The motivation comes from units in integral group rings and this new algorithm allows to handle unit groups of orders in $2$-by-$2$ matrices over rational quaternion algebras. The rings investigated are part of the so-called exceptional components of a rational group algebra.