A generalization of a theorem of Poincaré

Abstract:

Let $G$ be a finitely generated Fuchsian group of the first kind. Let $\varphi$ be a cusp form, and $f$ a solution to ${\theta _2}f = \varphi$, where ${\theta _2}$ is the Schwarzian derivative. Then for every $A \in G$ there is a Möbius transformation $\chi (A)$ such that $f \circ A = \chi (A) \circ f$. We show that the homomorphism $\chi$ from $G$ to Möbius transformations determines $\varphi$. The theorem for the special case where $G$ is the covering group of a compact surface was first proved by Poincaré.