On the resolvent of the Laplacian on functions for degenerating surfaces of finite geometry

Abstract

We consider families of degenerating hyperbolic surfaces. The surfaces are geometrically finite of fixed topological type. Let Z(s) be the Selberg Zeta function of a surface, and let Z_d(s) be the contribution of the pinched geodesics to the Zeta function. Extending a result of Hejhal and Wolpert, we prove that the quotient of these two terms converges to the Zeta function of the limit surface for all arguments s with re(s)>1/2. The technique is an examination of resolvent of the Laplacian, which is composed from that for elementary surfaces via meromorphic Fredholm theory. The resolvent is shown to converge on the complement of the essential spectrum of the limit surface. We also use this property to define approximate Eisenstein functions and scattering matrices.