Existence and Non-Existence of Torsion in Maximal Arithmetic Fuchsian Groups

Abstract

In [1], Borel discussed discrete arithmetic groups arising from quaternion algebras over number fields with particular reference to arithmetic Kleinian and arithmetic Fuchsian groups. In these cases, he described, in each commensurability class, a class of groups which contains all maximal groups. Developing results on embedding commutative orders of the defining number field into maximal or Eichler orders in the defining quaternion algebra, Chinburg and Friedman [2] stated necessary and sufficient conditions for the existence of torsion in this class of groups in terms of the defining arithmetic data. This was more fully explored in the case of Kleinian groups in [3]. In the case of Fuchsian groups, these results on the existence of torsion were extended to obtain formulas for the number of conjugacy classes of finite cyclic subgroups for each group in this class [8, 9]. In this paper, we examine, across the range of arithmetic Fuchsian groups, how widespread torsion is in maximal Fuchsian groups. Some studies in low genus cases (see e.g. [7, 12]) indicate that 2-torsion is very prevalent. The results obtained here substantiate that but we will also obtain maximal arithmetic Fuchsian groups which are torsion-free. The author is grateful to Alan Reid for conversations on parts of this paper.