Cusp forms for locally symmetric spaces of infinite volume

Let $G$ be a real simple linear connected Lie group of real rank one. Then, $X:=G\mathrm{/}K$ 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 $\mathcal{C}(\mathrm{\Gamma }\mathrm{\backslash }G)$ on $ \Gamma \backslash G $fortorsion-freegeometricallyfinitesubgroups$ \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 $L^2(\mathrm{\Gamma }\mathrm{\backslash }G)$ and that the right translation by elements of $G$ defines a representation on $\mathcal{C}(\mathrm{\Gamma }\mathrm{\backslash }G)$. 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 $\mathcal{C}(\mathrm{\Gamma }\mathrm{\backslash }G)$. It consists of the Schwartz functions which have vanishing “constant term” along the ordinary set ${\mathrm{\Omega }}_{\mathrm{\Gamma }}\subset \mathrm{\partial }X$ and along every cusp. We show that these two constant terms are in fact related by a limit formula if the cusp is of smaller rank (not of full rank). The main result of this thesis consists in proving a direct sum decomposition of the closure of the space of cusp forms in $L^2(\mathrm{\Gamma }\mathrm{\backslash }G)$ which respects the Plancherel decomposition in the case where $\mathrm{\Gamma }$ is convex-cococompact and noncocompact. For technical reasons, we exclude here that $X$ is the Cayley hyperbolic plane.