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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. 

In this paper, we introduce a property of topological dynamical systems that we call finite dynamical complexity. For systems with this property, one can in principle compute the K-theory of the associated crossed product C*-algebra by splitting it up into simpler pieces and using the methods of controlled K-theory. The main part of the paper illustrates this idea by giving a new proof of the Baum-Connes conjecture for actions with finite dynamical complexity. We have tried to keep the paper as self-contained as possible: we hope the main part will be accessible to someone with the equivalent of a first course in operator K-theory. In particular, we do not assume prior knowledge of controlled K-theory, and use a new and concrete model for the Baum-Connes conjecture with coefficients that requires no bivariant K-theory to set up. 

In this paper, we introduce a property of topological dynamical systems that we call finite dynamical complexity. For systems with this property, one can in principle compute the K-theory of the associated crossed product C*-algebra by splitting it up into simpler pieces and using the methods of controlled K-theory. The main part of the paper illustrates this idea by giving a new proof of the Baum-Connes conjecture for actions with finite dynamical complexity. We have tried to keep the paper as self-contained as possible: we hope the main part will be accessible to someone with the equivalent of a first course in operator K-theory. In particular, we do not assume prior knowledge of controlled K-theory, and use a new and concrete model for the Baum-Connes conjecture with coefficients that requires no bivariant K-theory to set up. 

The universal coefficient theorem is a fundamental problem in the classification theory for C*-algebras. In this article we develop a quantitative K-homology theory to prove the universal coefficient theorem for C*-algebras with finite complexity. 

The relative Novikov conjecture states that the relative higher signatures of manifolds with boundary are invariant under orientation-preserving homotopy equivalences of pairs. The relative Baum–Connes assembly encodes information about the relative higher index of elliptic operators on manifolds with boundary. In this paper, we study the relative Baum– Connes assembly map for any pair of groups and apply it to solve the relative Novikov conjecture when the groups satisfy certain geometric conditions.