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Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group. (English) Zbl 0969.11019
In a conference talk at Warwick in 1993, 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.

11F75 Cohomology of arithmetic groups
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
22E40 Discrete subgroups of Lie groups
Zbl 0851.22012
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