Introduction

This article is an extended version of the lecture given at UNED in July 1998 on “Topics on Riemann surfaces and Fuchsian groups” to mark the 25th anniversary of UNED.

The object of that lecture was to motivate the definition of arithmetic Fuchsian groups from the special and very familiar example of the classical modular group. This motivation proceeded via quaternion algebras and the lecture ended with the definition of arithmetic Fuchsian groups in these terms. This essay will go a little beyond that to indicate how the number theoretic data defining an arithmetic Fuchsian group can be used to determine geometric and group-theoretic information. No effort is made here to investigate other approaches to arithmetic Fuchsian groups via quadratic forms or to discuss and locate these groups in the general theory of discrete arithmetic subgroups of semi-simple Lie groups. Thus the horizons of this article are limited to giving one method of introducing an audience familiar with the ideas of Fuchsian groups and Riemann surfaces to the interesting special subclass of arithmetic Fuchsian groups.

Basics

A Fuchsian group is a discrete subgroup of SL(2,ℝ) or of PSL(2,ℝ) = SL(2,ℝ)/ < −I >. We will frequently employ the usual abuse of notation by writing elements of PSL(2,ℝ) as matrices, while strictly they are only determined up to sign.