Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group

Ulrich Bunke and Martin Olbrich

Review from Zentralblatt MATH:

In a conference talk at Warwick in 1993, {\it S. J. Patterson} formulated the conjecture that the divisor of the Selberg zeta-function 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 zeta-function at a given complex number as a higher Euler characteristic of that group cohomology.

In [ J. Reine Angew. Math. 467, 199-219 (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 zeta-function; higher Euler characteristic; group cohomology; invariant distribution

Classification (MSC2000): 11F75 11M36 37C30 22E40

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