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.

我们证明了诺维科夫猜想对于任何在允许的希尔伯特-哈达玛空间上允许等距和度量适当作用的离散组成立。允许的希尔伯特-哈达玛空间是一类（可能是无穷大的）非正弯曲的度量空间，其中包含与黎曼流形等轴测的密闭凸子集的密集序列。允许的希尔伯特-哈达玛空间的例子包括希尔伯特空间，某些简单连接且非正弯曲的黎曼-希尔伯特流形和无限维对称空间。因此，对于在完整，简单连接且非正弯曲的流形上正确且等距地作用的组，我们的主要定理可以看作是Novikov猜想上的Kasparov定理的无穷大类比。作为结果，我们表明，诺维科夫猜想对于一个封闭的光滑流形的体积保持微分集群的几何离散子集成立。这个结果受到康纳斯定理的启发，诺维科夫猜想持有与微分同构群的Gelfand-Fuchs类相关的更高签名。