Arithmetic Fuchsian Groups of Genus Zero

Abstract

If i is a flnite co-area Fuchsian group acting on H 2 , then the quotient H 2 =i is a hyperbolic 2-orbifold, with underlying space an orientable surface (possibly with punctures) and a flnite number of cone points. Through their close connections with number theory and the theory of automorphic forms, arithmetic Fuchsian groups form a widely studied and interesting subclass of flnite co-area Fuchsian groups. This paper is concerned with the distribution of arithmetic Fuchsian groups i for which the underlying surface of the orbifold H 2 =i is of genus zero; for short we say i is of genus zero. The motivation for the study of these groups comes from many difierent view