On Discontinuous Groups Operating on the Product of the Upper Half Planes

Let H be the direct product of the n upper half planes, and let be the connected component of the identity of the group of all analytic automorphisms of Hn. is the direct product of n subgroups G1, G2,. . . * Gq each of which is isomorphic to the group of all analytic automorphisms of the upper half plane H. Consider a discrete subgroup of such that the factor space 17 is of finite measure. (The notations Hn, or will keep, unless otherwise stated, these meanings throughout this paper. ) For the study of the groups of this type, it is important to investigate the case where is an irreducible subgroup of in the following sense. A discrete subgroup of is said to be irreducible if is not commensurable' with any direct product F' x F, where F' and F are respectively discrete subgroups of the partial products G' and G of =G, x G2 x. . . x Ga with = G' x G, G' I 1, G # 1. The main purpose of this paper was originally to calculate the dimension of the space of cusp foruis for an irreducible group by means of Selberg's trace formula; however, for the sake of completeness, and in view of the fact that no proof has been published as yet for the results stated by Pyatetzki-Shapiro in 6, it has been found desirable to prove these results here, following the ideas indicated by Pyatetzki-Shapiro himself. Therefore, no new results will be found in our? ? 1-3 except for some supplementary results such as Theorem 1, and for Theorems 6 and 7, which are proved under an additional condition on the fundamental domain of (Assumption (F) in No. 11,? 3). Theorems 2, 3, 4, 5 are the restatements of the results of Pyatetzki-Shapiro; but, for the sake of simplicity, we state here the latter three theorems only for irreducible case. For these results, Pyatetzki-Shapiro has indicated in 6 only a sketch of proof for Theorem 3 as well as the implications between them. However, our proofs will be probably more or less the same as the ones he has. In? ? 4-5, we shall calculate the dimension of the space of cusp forms for an irreducible group under the assumption (F). The main result is