Selberg Zeta function and hyperbolic Eisenstein series

Abstract

Let [Formula: see text] be a Fuchsian group acting on the hyperbolic upper half-plane [Formula: see text], such that [Formula: see text] is a geometrically finite Riemann surface with respect to the natural hyperbolic metric induced from [Formula: see text]. If [Formula: see text] is hyperbolic then following [J. Kudla and S. Millson, Harmonic differentials and closed geodesics on a Riemann surface, Invent. Math. 54 (1979) 193–211; J. D. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math. 293 (1977) 143–203], there is a corresponding hyperbolic Eisenstein series. In this paper, we study the limiting behavior of hyperbolic Eisenstein series on a degenerating family of geometrically finite hyperbolic surfaces. In particular, we give a partial lightening to a question of Ji, concerning the approximation of Eisenstein series during degeneration (see Proposition 5.2 and Theorem 5.2).