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Annals of Mathematics, II. Series Vol. 149, No. 2, pp. 627689 (1999) 

Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian groupUlrich Bunke and Martin OlbrichReview from Zentralblatt MATH: In a conference talk at Warwick in 1993, {\it S. J. Patterson} formulated the conjecture that the divisor of the Selberg zetafunction for a convex cocompact group $\Gamma$ should be expressible in terms of the $\Gamma$cohomology with coefficients in the module of distributions supported on the limit set of $\Gamma$. He gave a formula expressing the order of the zetafunction at a given complex number as a higher Euler characteristic of that group cohomology. In [ J. Reine Angew. Math. 467, 199219 (1995; Zbl 0851.22012)] the present authors proved this conjecture in the cocompact rank one case. The present paper contains the proof for the convex cocompact hyperbolic case, i.e. the setting of the original conjecture. The central tool is a canonical invariant extension of a given invariant distribution beyond the limit set. These distributions are defined on spaces of sections of vector bundles which are parametrized by a complex parameter $\lambda$. For $\text{Re}(\lambda)$ large the extension is gotten by summation which will converge then. The central achievement then is to extend this extension as a meromorphic function of $\lambda$ to the complex plane. Reviewed by Anton Deitmar Keywords: Patterson conjecture; convex cocompact hyperbolic group; divisor; Selberg zetafunction; higher Euler characteristic; group cohomology; invariant distribution Classification (MSC2000): 11F75 11M36 37C30 22E40 Full text of the article:
Electronic fulltext finalized on: 8 Sep 2001. This page was last modified: 21 Jan 2002.
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