Reid, Alan709158812008-08-282017-05-112008-08-282017-05-112005http://hdl.handle.net/2152/2274textWe 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.electronicengCopyright is held by the author. Presentation of this material on the Libraries' web site by University Libraries, The University of Texas at Austin was made possible under a limited license grant from the author who has retained all copyrights in the works.Fuchsian groupsThesis