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Fuchsian groups of the second kind and representations carried by the limit set. (English) Zbl 0880.30035

The authors study the cohomology of a Fuchsian group of the second kind with coefficients in certain infinite dimensional representations. Namely, one considers a nontrivial discrete subgroup \(\Gamma\) of the group \(G=SL(2,{\mathbb{R}})\), acting freely on the real hyperbolic plane \(X=G/S^1\). Denote by \(\Lambda\) the limit set of an arbitrary orbit \(\Gamma x\), and let \(\lambda\) be a complex number that parametrizes the principal series representation \(H^{\lambda}\) of \(G\) on a Hilbert space (defined in the paper). Then \(H^{\lambda}_{-\omega}\) will denote the space of its hyperfunction vectors, while \(H^{\lambda}_{-\omega, \Lambda}\) the subspace of hyperfunction sections with support in the limit set \(\Lambda\). The aim of the paper is to study the cohomology groups \(H^{*}(\Gamma,H^{\lambda}_{-\omega, \Lambda})\). The interest in these cohomology groups is motivated by a conjecture that relates their dimensions with the order of the singularities of the Selberg zeta function associated to \(\Gamma\).

MSC:

30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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