A groupoid approach to the equivariant coarse Baum--Connes conjecture
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911628941852672 |
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| author | Guo, Liang |
| author_facet | Guo, Liang |
| contents | In this paper, we develop a groupoid approach to the equivariant coarse Baum--Connes conjecture. For a bounded geometry metric space $X$ equipped with a proper, free, and isometric action of a countable discrete group $Γ$, we introduce the equivariant coarse groupoid $G(X, Γ)$. We prove that the groupoid Baum--Connes conjecture for $G(X, Γ)$ with coefficients in $\ell^{\infty}(X,\mathcal{K})^Γ$ is equivalent to the equivariant coarse Baum--Connes conjecture for $(X, Γ)$ using a localization algebra description of equivariant $KK^\mathcal{G}$-theory for étale groupoids. As applications of this framework, we prove that if the space $X$ admits a coarse embedding into Hilbert space (which is not required to be $Γ$-equivariant), then the equivariant coarse Novikov conjecture holds for $(X, Γ)$, i.e., the assembly map $μ_{X,Γ}$ is an injection. We also obtain a new proof of the equivariant coarse Baum--Connes conjecture if $X$ admits an equivariant coarse embedding into Hilbert space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25595 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A groupoid approach to the equivariant coarse Baum--Connes conjecture Guo, Liang K-Theory and Homology Metric Geometry Operator Algebras 58J22 (Primary), 22A22, 51F30, 19K35 In this paper, we develop a groupoid approach to the equivariant coarse Baum--Connes conjecture. For a bounded geometry metric space $X$ equipped with a proper, free, and isometric action of a countable discrete group $Γ$, we introduce the equivariant coarse groupoid $G(X, Γ)$. We prove that the groupoid Baum--Connes conjecture for $G(X, Γ)$ with coefficients in $\ell^{\infty}(X,\mathcal{K})^Γ$ is equivalent to the equivariant coarse Baum--Connes conjecture for $(X, Γ)$ using a localization algebra description of equivariant $KK^\mathcal{G}$-theory for étale groupoids. As applications of this framework, we prove that if the space $X$ admits a coarse embedding into Hilbert space (which is not required to be $Γ$-equivariant), then the equivariant coarse Novikov conjecture holds for $(X, Γ)$, i.e., the assembly map $μ_{X,Γ}$ is an injection. We also obtain a new proof of the equivariant coarse Baum--Connes conjecture if $X$ admits an equivariant coarse embedding into Hilbert space. |
| title | A groupoid approach to the equivariant coarse Baum--Connes conjecture |
| topic | K-Theory and Homology Metric Geometry Operator Algebras 58J22 (Primary), 22A22, 51F30, 19K35 |
| url | https://arxiv.org/abs/2604.25595 |