Extension of a key identity

Abstract

In this article, we extend a certain key identity proved by J. Jorgenson and J. Kramer for compact hyperbolic Riemann surfaces to noncompact hyperbolic Riemann orbisurfaces of finite volume, which can be realized as the quotient space of the action of a Fuchsian subgroup of the first kind acting on the hyperbolic upper half-plane. The key identity of J. Jorgenson and J. Kramer relates the two natural metrics, namely the hyperbolic metric and the canonical metric defined on a hyperbolic Riemann surface. Via the spectral expansion of the hyperbolic heat kernel, this identity serves as a trace formula relating the weight 2 cusp forms with Maass forms. Our result is an extension of the key identity to elliptic fixed points and cusps at the level of currents acting on a certain space of singular functions. Our result serves as the starting point for extending the work of J. Jorgenson and J. Kramer to non compact hyperbolic Riemann orbisurfaces. In particular, to the problem of deriving bounds for the canonical Green's function defined on a noncompact hyperbolic Riemann orbisurface of finite volume, which is being addressed in an upcoming article by the same author.