The Novikov Conjecture, the Group of Volume Preserving Diffeomorphisms and Non-Positively Curved Hilbert Manifolds

Abstract

We prove that the Novikov conjecture is satisfied by any discrete torsion-free group admitting an isometric and metrically proper action on a non-positively curved complete simply-connected Hilbert manifold with enough finite-dimensional complete geodesic submanifolds, under the assumption that this isometric action can be continuously deformed to the trivial action. The proof makes use of a C*-algebra associated to a non-positively curved simply-connected Hilbert manifold that generalizes a construction of Higson and Kasparov and parallels a construction of Kasparov and Yu.