Resolvent of the Laplacian on geometrically finite hyperbolic manifolds (Q663297)

Let \(X=\Gamma \backslash \mathbb{H}^{n+1}\) be an hyperbolic manifold and denote by \(\Omega_{\Gamma}\) the domain of discontinuity on which the group \(\Gamma\) acts. Then \(X\) is called \textit{geometrically finite hyperbolic} if \(\Gamma \backslash (\mathbb{H}^{n+1}\cup \Omega_{\Gamma})\) can be decomposed into the union of a compact set and a finite number of standard cusp regions. In this setting the authors prove the existence of a meromorphic extension of the resolvent \(R_X(s)\) of the Laplacian from the region \(\text{Re }(s)> n/2\) to the complex plane. Moreover, they show that \(R_X(s)\) is bounded as an operator on appropriate weighted \(L^2\)-spaces. Then in a second description the resolvent is expressed by summing translates of the free space resolvent \(R_{\mathbb{H}{n+1}}(s)\) by elements \(\gamma\in \Gamma\). The resulting expression converges for \(\text{Re}(s) >(n-1)/2\) and leads to finer estimates of the resolvent kernel close to the critical line \(\text{Re }(s)=n/2\). Interesting applications of these results to the meromorphic extension of the Poincaré series for the group \(\Gamma\) and a relation to lattice point counting are given. The final part of the paper is concerned with the scattering theory on \(X\). The Poisson operator \(P_X(s)\) is defined, its meromorphic continuation for \(s\in \mathbb{C}\) as well as some mapping properties are shown. Then the authors comment on various relations between the Poisson and the scattering operator.