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

We consider families (Yn) of degenerating hyperbolic surfaces. The surfaces are geometrically finite of fixed topological type. Let Zn be the Selberg Zeta function of Yn, and let zn be the contribution of the pinched geodesics to Zn. Extending a result of Wolpert’s, we prove that Zn(s)/zn(s) converges to the Zeta function of the limit surface if 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 ( n − t)−1 is shown to converge for all t /∈ [1/4,∞). We also use this property to define approximate Eisenstein functions and scattering matrices. © 2006 Elsevier Inc. All rights reserved.