Zusammenfassung
We consider families (Y-n) of degenerating hyperbolic surfaces. The surfaces are geometrically finite of fixed topological type. Let Z(n) be the Selberg Zeta function of Y-n, and let z(n) be the contribution of the pinched geodesics to Z(n) Extending a result of Wolpert's, we prove that Z(n)(s)/z(n)(s) converges to the Zeta function of the limit surface if Re(s) > 1/2. The technique is an ...
Zusammenfassung
We consider families (Y-n) of degenerating hyperbolic surfaces. The surfaces are geometrically finite of fixed topological type. Let Z(n) be the Selberg Zeta function of Y-n, and let z(n) be the contribution of the pinched geodesics to Z(n) Extending a result of Wolpert's, we prove that Z(n)(s)/z(n)(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 (Delta n - t)(-1) is shown to converge for all t is not an element of [14, infinity). We also use this property to define approximate Eisenstein functions and scattering matrices. (c) 2006 Elsevier Inc. All rights reserved.
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