From the introduction: Let us consider hyperbolic manifolds \(\Gamma\setminus \mathbb{H}^{n+1}\), where \(\Gamma\) is a discrete group acting on the \((n+1)\)-dimensional hyperbolic space \(\mathbb{H}^{n+1}\). Then the spectrum of the structure, that is, the spectrum of the invariant Laplacian which acts on automorphic functions with respect to \(\Gamma\), depends on both \(\Gamma\) and \(\mathbb{H}^{n+1}\). Let us denote it by \(\text{Sp} (\Gamma;\mathbb{H}^{n+1})\). Embedding \(\Gamma\) into a higher-dimensional hyperbolic space \(\mathbb{H}^{m+1}\), \(m>n\), one has that \(\text{Sp}(\Gamma;\mathbb{H}^{n+1})\) is very different from \(\text{Sp} (\Gamma; \mathbb{H}^{m+1})\), despite the fact that \(\Gamma\) is kept fixed.
We use this phenomenon in order to give a simple proof of the analytic continuation of the resolvent kernel for a convex cocompact Kleinian group. This is known to be a difficult problem. The idea goes back to the program concerning the analytic continuation of the Eisenstein series for Kleinian groups, but has never been published before.
For a different method of the analytic continuation of the resolvent kernel see R. R. Mazzeo and R. B. Melrose [J. Funct. Anal. 75, 260–310 (1987; Zbl 0636.58034)]. The main difference which exists between Mazzeo-Melrose’s theory and the one given here, at least as far as the analytic continuation of the Eisenstein series is concerned, lies in the fact that they cannot produce the functional equation for the Eisenstein series as they do not possess the “Maass-Selberg” relations.