Derived Arithmetic Fuchsian Groups of Genus Two

TL;DR

This paper classifies all torsion-free derived arithmetic Fuchsian groups of genus two, establishes their relation to quaternion algebras, and provides explicit generators for these groups using computational methods.

Contribution

It provides a complete classification of such groups by commensurability class and explores their quaternion algebra origins, including explicit generator determination.

Findings

No such groups from quaternion algebras over fields of degree > 5

Existence of maximal orders in related quaternion algebras

Explicit generators for several examples

Abstract

We classify all torsion-free derived arithmetic Fuchsian groups of genus two by commensurability class. In particular, we show that there exist no such groups arising from quaternion algebras over number fields of degree greater than 5. We also prove some results on the existence and form of maximal orders for a class of quaternion algebras related to these groups. Using these results in conjunction with a computer program, one can determine an explicit set of generators for each derived arithmetic Fuchsian group containing a torsion-free subgroup of genus two. We show this for a number of examples.