Cohomology of Fuchsian Groups

In his paper 'Remarks on the cohomology of groups' [11], Andre Weil computes the dimension of the space of cohomology classes of degree one given by parabolic cocycles of a group T acting discontinuously on the upper half plane H such that the quotient is compactifiable. In the particular case when T does not have fixed points in H and H/V is compact, this has also been done by M. S. Narasimhan and C. S. Seshadri in [6]. Whereas Weil used purely group-theoretic methods in his proof, the methods of [6] are based on the cohomology of local systems. The purpose of the present article is to give a proof of Weil's formula in the general case on the lines of [6] by using cohomology theory of sheaves. We make particular use of the Tohoku paper of Grothendieck [3], to which we shall refer in the sequel as 'Tohoku'.