Fundamental Domains for Shimura Curves

We describe a process for defining and computing a fundamental domain in the upper half plane H of a Shimura curve X 0 (N) associated to an order in a quaternion algebra A/Q. A fundamental domain for X 0 (N) realizes a finite presentation of the quaternion unit group, modulo units of its center. We give explicit examples of domains for the curves X 0 (1), X 0 (1), and X 35 0 (1). The first example is a classical example of a triangle group and the second is a corrected version of that appearing in the book of Vignéras [13], due to Michon. These examples are also treated in the thesis of Alsina [1]. The final example is new and provides a demonstration of methods to apply when the group action has no elliptic points.