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The spectral theory of geometrically periodic hyperbolic 3-manifolds. (English) Zbl 0584.58047

Mem. Am. Math. Soc. 335, 161 p. (1985).
The purpose of this memoir is to study the spectral theory of certain Kleinian groups \(\Gamma\) which are not geometrically finite. The groups in question are ones for which there exists a geometrically finite group \(\Gamma^*\) of finite covolume so that \(\Gamma\) is normal in \(\Gamma^*\) and \(\Gamma^*/\Gamma\) is infinite cyclic. The author uses the spectral theory of \(\Gamma^*\) and the perturbation theory of elliptic operators to obtain a very precise description of the spectrum of \(\Gamma\). As a rather striking application of his theory the author obtains asymptotic formulae for the ”lattice point problem” for \(\Gamma\) and for the counting function for the lengths of closed geodesics. This is the first time that such results have been obtained for non- geometrically finite groups.
Reviewer: S.J.Patterson

MSC:

58J70 Invariance and symmetry properties for PDEs on manifolds
11F85 \(p\)-adic theory, local fields
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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