The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces
The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces
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DOI：
10.1007/s00039-021-00563-7
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发表时间：
2018-11
影响因子：
2.2
通讯作者：
Sherry Gong;Jianchao Wu;Guoliang Yu
中科院分区：
文献类型：
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作者：
Sherry Gong;Jianchao Wu;Guoliang Yu
我们证明了Novikov猜想对任何在可容许的Hilbert-Hadamard空间上允许等距度量真作用的离散群都成立。容许Hilbert-Hadamard空间是一类(可能是无限维的)非正曲度量空间，它包含与黎曼流形等距的闭凸子集的稠密序列。允许的Hilbert-Hadamard空间的例子包括Hilbert空间、某些单连通和非正弯曲的Riemannian-Hilbertian流形和无限维对称空间。因此，我们的主要定理可以看作是Kasparov关于Novikov猜想上的Kasparov定理的无限维类似，该猜想适用于完全、单连通和非正曲线流形上适当且等距作用的群。因此，我们证明了Novikov猜想对闭光滑流形的保体积微分同胚群的几何离散子群成立。这一结果是受Connes定理的启发，Novikov猜想适用于与Gelfand-Fuchs类的不同态射群有关的高签名。
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.