| Abstract: | In this paper, the Yang-Yau inequality for the first eigenvalue of the Laplace operator on a compact Riemann surface is carried to the case of Fuchsian groups of the first kind. With its help, for specific subgroups of the modular group PSL(2, ), the existence of cuspidal representations of the complementary series in the decomposition of regular representations of the group PSL(2, ) into irreducible components is proved. In addition, a lower bound for the degree of an arbitrary nonconstant meromorphic function, automorphic with respect to some congruence subgroup  of PSL(2,) is given in terms of the index of in PSL (2,) only.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vo!o 134, pp. 157–168, 1984o |
