Cohomology of discontinuous groups
Let \(\Delta\) be a discontinuous group of dianalytic automorphisms of the upper half plane H and let \(\Gamma\) be the subgroup consisting of the analytic automorphisms in \(\Delta\) ; further, assume \(\Gamma\) \(\neq \Delta\). Then \(\Delta\) operates on \(H^+=H\cup \{parabolic\) fixed points of \(\Gamma\) \(\}\) and \(Y=H^+/\Delta\) is a non-orientable Klein surface. Now let \(\tau\) be a representation of \(\Delta\) on a finite-dimensional complex vector space W and let \({\mathcal F}\) be the sheaf of \(\Delta\)- invariants of the direct image sheaf on Y of the constant sheaf on \(H^+\) with fibre W. Assuming that Y is compact and has no boundary, the author identifies \(H^ 1(Y,{\mathcal F})\) with a subspace of \(H^ 1(\Delta,\tau)\) and computes the Euler characteristic of \({\mathcal F}\) in terms of the genus of Y, the dimension of W, and the rank of some endomorphisms of W of the form id-\(\tau\) (c), where c runs through a certain set of elements of finite order in \(\Gamma\). The proof is analogous to a similar result of the same author on Fuchsian groups [cf. J. Ind. Math. Soc., Nov. Ser. 38(1974), 51-63 (1975; Zbl 0381.20035)].
scientific article; zbMATH DE number 3888029
scientific article