Cohomology of discontinuous groups
type: Item
instance of: scholarly article
title: Cohomology of discontinuous groups
zbMATH Open document ID: 0557.30039
DOI: 10.1007/BF02861834
author: P. Krishna Prasad
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1984-01-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 30F40
zbMATH DE Number: 3888029
zbMATH Keywords: cohomology
zbMATH Keywords: Klein surface
zbMATH Keywords: sheaf
zbMATH Keywords: Euler characteristic
zbMATH Keywords: Fuchsian groups
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02861834
OpenAlex ID: W2013264713
description: scientific article; zbMATH DE number 3888029