A note on "On the appearance of Eisenstein series through degeneration"

Abstract

Let $M = \Gamma \backslash H$ be a geometrically finite hyperbolic surface, realized as the quotient of the hyperbolic upper half plane $H$ by a geometrically finite discrete group of isometries acting on $H$. To a parabolic element of the uniformizing group $\Gamma$, there is an associated 1-form parabolic Eisenstein series. To a primitive hyperbolic element, then, following ideas due to Kudla-Millson, there is a corresponding 1-form hyperbolic Eisenstein series. In this article, we study the limiting behavior of these hyperbolic Eisenstein series on a degenerating family of hyperbolic Riemann surfaces of finite type, using basically the limiting behavior of counting functions associated to degenerating hyperbolic Riemann surfaces. In this sense, we generalize the results obtained in Garbin, Jorgenson and Munn (Comment Math Helv 83:701-721, 2008) to the case of geometrically finite hyperbolic surfaces of infinite volume and form-valued parabolic and hyperbolic Eisenstein series.