The Equivariant Coarse Novikov Conjecture and Coarse Embedding
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Communications in
Mathematical Physics
The Equivariant Coarse Novikov Conjecture and Coarse Embedding Benyin Fu1 , Xianjin Wang2 , Guoliang Yu3 1 College of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance,
Shanghai 201209, People’s Republic of China. E-mail: [email protected]
2 College of Mathematics and Statistics, Chongqing University (at Huxi Campus), Chongqing 401331,
People’s Republic of China. E-mail: [email protected]
3 Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA.
E-mail: [email protected] Received: 14 September 2019 / Accepted: 24 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: The equivariant coarse Novikov conjecture provides an algorithm for determining nonvanishing of equivariant higher index of elliptic differential operators on noncompact manifolds. In this article, we prove the equivariant coarse Novikov conjecture under certain coarse embeddability conditions. More precisely, if a discrete group acts on a bounded geometric space X properly, isometrically, and with bounded distortion, X/ and admit coarse embeddings into Hilbert space, then the -equivariant coarse Novikov conjecture holds for X . Here bounded distortion means that for any γ ∈ , supx∈Y d(γ x, x) < ∞, where Y is a fundamental domain of the -action on X . 1. Introduction Let X be a proper metric space, let be a countable discrete group. Assume that acts on X properly and isometrically. One can define an equivariant higher index map (see [4,31,33]) I nd : lim K ∗ (Pd (X )) → K ∗ (C ∗ (X ) ), d→∞
where K ∗ (Pd (X )) is the -equivariant K -homology group of Rips complex Pd (X ) and C ∗ (X ) is the equivariant Roe algebra, K ∗ (C ∗ (X ) ) is the receptacle of higher index for elliptic differential operators. The equivariant coarse Novikov conjecture states that the equivariant higher index map is injective. This conjecture provides an algorithm for determining the nonvanishing of equivariant higher index of elliptic differential operators. Nonvanishing of equivariant higher index has important applications to geometry and topology such as the positive scalar curvature problem (see [12,13,16,20,28,35]). When the -action is cocompact, i.e., X/ is compact, C ∗ (X ) is Morita equivalent Benyin Fu is supported by NSFC (Nos. 11871342, 11771143). Xianjin Wang is supported by NSFC (No. 11771061). Guoliang Yu is supported by NSF (Nos. 1564398, 1700021) and NSFC (No. 11420101001).
B. Fu, X. Wang, G. Yu
to Cr∗ (), the reduced C ∗ -algebra of , then the equivariant higher index map is the Baum–Connes map introduced by Baum, Connes and Higson (see [1,2]). When is trivial, the equivariant higher index map is the coarse Baum–Connes map introduced by Roe, Higson and Yu (see [5,8–10,20,26,27,37]). Gromov [14] introduced the following definition of coarse embedding. Definition 1.1. Let X and Y be two metric spaces. A map f : X → Y is said to be a coarse embedding if there exist non-decreasing functio
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