Let \(\Gamma\) be a geometric finite discrete group of isometries of the hyperbolic space \(H^{n+1}\); this means that \(\Gamma\) admits a fundamental domain with finitely many sides. By passing to a subgroup of finite index, one may assume \(\Gamma\) is torsion free. Let \(X:=H^{n+1}/\Gamma\) be the associated orbit space equipped with the complete hyperbolic metric of constant sectional curvature \(-1\). Assume \(X\) has infinite hyperbolic volume; also assume \(\Gamma\) has no parabolic subgroup with irrational rotations. Then the Laplace-Beltrami operator \(\Delta_X\) has at most a finite number of discrete eigenvalues in the interval \([0,{n\over 2}^2)\) and has continuous spectrum in \([{n\over 2}^2,\infty)\).
The authors construct a parametrix for the resolvent kernel of the Laplace operator and study the meromorphic continuation through the spectrum.
For the entire collection see [Zbl 0922.00026].