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/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 C(Г\G) on Г\G for torsion-free geometrically finite subgroups Г of G. We show that it has a Fréchet space structure, that the space of compactly supported smooth functions is dense in this space, that it is contained in L^2(Г\G) and that the right translation by elements of G defines a representation on C(Г\G). Moreover, we define the space of cusp forms °C(Г\G) on Г\G, which is a geometrically defined subspace of C(Г\G). It consists of the Schwartz functions which have vanishing ''constant term'' along the ordinary set Ω ⊂ ∂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(Г\G) which respects the Plancherel decomposition in the case where Г is convex-cococompact and noncocompact. For technical reasons, we exclude here that X is the Cayley hyperbolic plane.