The rational strong Novikov conjecture is a deep problem in noncommutative geometry. It implies important conjectures in manifold topology and differential geometry such as the (classical) Novikov conjecture on higher signatures and the Gromov-Lawson conjecture on positive scalar curvature. Using C*-algebraic and K-theoretic tools, we prove that the rational strong Novikov conjecture holds for any discrete group admitting an isometric and proper action on an admissible Hilbert-Hadamard space, which is a (typically infinite-dimensional) generalization of complete simply connected nonpositively curved Riemannian manifolds. In particular, this result applies to geometrically discrete subgroups of the group of volume preserving diffeomorphisms of a closed smooth manifold. This is joint work with Sherry Gong and Guoliang Yu.