  We prove that the Novikov conjecture holds for any discrete group admitting
an isometric and metrically proper action on an admissible Hilbert-Hadamard
space. Admissible Hilbert-Hadamard spaces are a class of (possibly
infinite-dimensional) non-positively curved metric spaces that contain dense
sequences of closed convex subsets isometric to Riemannian manifolds. Examples
of admissible Hilbert-Hadamard spaces include Hilbert spaces, certain simply
connected and non-positively curved Riemannian-Hilbertian manifolds and
infinite\-/dimensional symmetric spaces. Thus our main theorem can be
considered as an infinite-dimensional analogue of Kasparov's theorem on the
Novikov conjecture for groups acting properly and isometrically on complete,
simply connected and non-positively curved manifolds. As a consequence, we show
that the Novikov conjecture holds for geometrically discrete subgroups of the
group of volume preserving diffeomorphisms of a closed smooth manifold. This
result is inspired by Connes' theorem that the Novikov conjecture holds for
higher signatures associated to the Gelfand-Fuchs classes of groups of
diffeormorphisms.
