Derived arithmetic Fuchsian groups of genus two

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 certain class of quaternion algebras. These can in turn be used to find an explicit set of generators for each derived arithmetic group containing a torsion-free subgroup of genus two. We show this for a number of examples.