## Oliver Club

We consider complete analytic Riemannian manifolds of bounded

nonpositive curvature. These include locally symmetric and, in

particular, arithmetic manifolds, the study of which is strongly

related to the theory of lattices and arithmetic groups. The

complexity of such manifolds is controlled by the volume. This

phenomenon can be measured in terms of the growth of topological,

geometric, algebraic, arithmetic and representation theoretic

invariants, such as Betti numbers and torsion, optimal presentations

of $\Gamma=\pi_1(M)$, the minimal size of a triangulation as well as

invariants related to the Plancherel measure associated to

$L_2(G/\Gamma)$. Other problems concern the number of manifolds of a

certain type and bounded volume. In the talk I will give an overview

of the theory.