| Let K be a number field with r real places and s complex places, and let OK be the ring of integers of K. The group PSL 2( OK ) embeds discretely in PSL2( R )r x PSL2( C )s, the group of orientation preserving isometrics of Xr,s = [ H2 ]r x [ H3 ]s, and acts with finite covolume. Hence for any finite index subgroup, Gamma of PSL2( OK ), the quotient [ H2 ]r x [ H3 ]s/Gamma is a finite volume (2 r + 3s)-dimensional orbifold. The quotient Xr,s/PSL2( OK ) has hK cusps, where hK is the class number of OK , therefore the quotient by Gamma has at least hK cusps. Petersson proved that there are only finitely many congruence subgroups of PSL2( Z ) whose quotient has one cusp. We show that when K is an imaginary quadratic there are only finitely many maximal congruence subgroups whose quotient has one cusp. In contrast, under the assumption of the Generalized Riemann Hypothesis, we show that if K is neither Q nor an imaginary quadratic, and i ∉ K then there are infinitely many maximal congruence subgroups whose quotient has one cusp, relating this condition to a generalization of Artin's primitive root conjecture. |
