In this paper, we consider the problem: what kinds of Riemann surfaces S will have rst eigenvalue 1 (S) large? Bye large, we mean that there is some positive lower bound for S independent of the genus. There are a number of constructions of surfaces with 1 large, the main example being the quotients of the hyperbolic plane by the congruence subgroups of PSL(2; Z). Most of the known examples rel...