The Equivariant Coarse Novikov Conjecture and Coarse Embedding

Fu, Benyin; Wang, Xianjin; Yu, Guoliang

doi:10.1007/s00220-020-03754-9

Cited by 6 publications

(14 citation statements)

References 31 publications

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“…Two effective sources of index-theoretic obstructions to metrics of positive scalar curvature on noncompact manifolds are coarse index theory and Callias-type index theory. For some results involving coarse index theory, see for example [38] and the literature on the coarse Novikov conjecture, in particular [12] for the equivariant setting we are interested in here. We will use Callias-type index theory.…”

Section: Introduction Results On Positive Scalar Curvature a Callias-...mentioning

confidence: 99%

Positive Scalar Curvature and Callias-Type Index Theorems for Proper Actions

Guo

2019

Bull. Aust. Math. Soc.

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For a proper action by a locally compact group G on a manifold M with a G-equivariant Spin-structure, we obtain obstructions to the existence of complete G-invariant Riemannian metrics with uniformly positive scalar curvature. We focus on the case where M/G is noncompact. The obstructions follow from a Callias-type index theorem, and relate to positive scalar curvature near hypersurfaces in M . We also deduce some other applications of this index theorem. If G is a connected Lie group, then the obstructions to positive scalar curvature vanish under a mild assumption on the action. In that case, we generalise a construction by Lawson and Yau to obtain complete G-invariant Riemannian metrics with uniformly positive scalar curvature, under an equivariant bounded geometry assumption.

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Section: Introduction Results On Positive Scalar Curvature a Callias-...mentioning

confidence: 99%

Positive Scalar Curvature and Callias-Type Index Theorems for Proper Actions

Guo

2019

Bull. Aust. Math. Soc.

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“…However, a recent ingenious example due to Arzhantseva and Tessera [1] illuminates that a metric space with a group action might not admit a coarse embedding into Hilbert space even if both the group and the quotient space are coarsely embeddable. Inspired by their example, the first author together with Wang and Yu [10] managed to show that the equivariant index map remains injective for group actions with bounded distortion when both the group and the quotient space admit coarse embeddings into Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

The equivariant coarse Baum-Connes conjecture for actions by a-T-menable groups

Fu¹,

Zhang²

2022

Preprint

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The equivariant coarse Baum-Connes conjecture was firstly introduced by Roe [29] as a unified way to approach both the Baum-Connes conjecture and its coarse counterpart. In this paper, we prove that if an a-T-menable group Γ acts properly and isometrically on a bounded geometry metric space X with controlled distortion such that the quotient space X/Γ is coarsely embeddable, then the equivariant coarse Baum-Connes conjecture holds for this action. This answers a question posed in [8] affirmatively.

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“…In [10], Fu and Wang showed that the equivariant coarse Baum-Connes conjecture holds for a Γ-space X with bounded geometry which admits an equivariant coarse embedding into Hilbert space. In [11], Fu, Wang and Yu proved that if a discrete group Γ acts properly and isometrically on a space X with bounded geometry, such that both X/Γ and Γ admit a coarse embedding into Hilbert space, and that the action has bounded distortion, then the equivariant higher index map is injective for the Γ-space X.…”

Section: Introductionmentioning

confidence: 99%

“…The reason why they do not coarsely embed into Hilbert relies on the fact that both groups contain a certain relative expanders [1]. If we take X = Z 2 ≀ G H or Z 2 ≀ G (H × F n ), and Γ = G Z 2 , then we come across with a situation that Γ is a torsion group, the Γ-action on X does not have bounded distortion, and that the space X does not even coarsely embed into Hilbert space, let alone admits a Γ-equivariant coarse embedding into Hilbert space, namely, a situation which does not satisfy the assumptions in each of the main results in [28,10,11] mentioned above.…”

Section: Introductionmentioning

confidence: 99%

The equivariant coarse Baum-Connes conjecture for metric spaces with proper group actions

Deng¹,

Fu²,

Wang³

2021

Preprint

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The equivariant coarse Baum-Connes conjecture interpolates between the Baum-Connes conjecture for a discrete group and the coarse Baum-Connes conjecture for a proper metric space. In this paper, we study this conjecture under certain assumptions. More precisely, assume that a countable discrete group Γ acts properly and isometrically on a discrete metric space X with bounded geometry, not necessarily cocompact. We show that if the quotient space X/Γ admits a coarse embedding into Hilbert space and Γ is amenable, and that the Γ-orbits in X are uniformly equivariantly coarsely equivalent to each other, then the equivariant coarse Baum-Connes conjecture holds for (X, Γ). Along the way, we prove a K-theoretic amenability statement for the Γ-space X under the same assumptions as above, namely, the canonical quotient map from the maximal equivariant Roe algebra of X to the reduced equivariant Roe algebra of X induces an isomorphism on K-theory.

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The Equivariant Coarse Novikov Conjecture and Coarse Embedding

Cited by 6 publications

References 31 publications

Positive Scalar Curvature and Callias-Type Index Theorems for Proper Actions

Positive Scalar Curvature and Callias-Type Index Theorems for Proper Actions

The equivariant coarse Baum-Connes conjecture for actions by a-T-menable groups

The equivariant coarse Baum-Connes conjecture for metric spaces with proper group actions

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