Let $ G $ be a real simple linear connected Lie group of real rank one. Then, $ X := G/K $ is a Riemannian symmetric space with strictly negative sectional curvature. By the classification of these spaces, $X$ is a real/complex/quaternionic hyperbolic space or the Cayley hyperbolic plane. We define the Schwartz space $ \mathscr{C}(\Gamma \backslash G) $ on $ \Gamma \backslash G $ for torsion-free geometrically finite subgroups $ \Gamma $ of $G$. We show that it has a Fr\'echet space structure, that the space of compactly supported smooth functions is dense in this space, that it is contained in $ L^2(\Gamma \backslash G) $ and that the right translation by elements of $G$ defines a representation on $ \mathscr{C}(\Gamma \backslash G) $. Moreover, we define the space of cusp forms $ \deg\mathscr{C}(\Gamma \backslash G) $ on $ \Gamma \backslash G $, which is a geometrically defined subspace of $ \mathscr{C}(\Gamma \backslash G) $. It consists of the Schwartz functions which have vanishin 查看全文>>