Rectified Baum-Connes conjecture for transformation groupoids

Let Γ\Gamma be a finitely generated discrete group acting on a metrizable finite-dimensional compact space XX, and let G=XΓ{\mathcal G}=X\rtimes\Gamma. Write RKi(Γ,X)RK_i(\Gamma,X) for the domain of the Baum-Connes map, let Cmax(G)C^*_{\max}({\mathcal G}) be the maximal crossed-product completion, and let Cϵ(G)=C(X)ϵΓC^*_{\epsilon}({\mathcal G})=C(X)\rtimes_{\epsilon}\Gamma be the Baum-Guentner-Willett minimal exact and Morita-compatible crossed product. For iZ2i\in\mathbb Z_2, let

μi,GBGW:RKi(Γ,X)Ki(Cϵ(G))\mu^{BGW}_{i,{\mathcal G}}:RK_i(\Gamma,X)\longrightarrow K_i(C^*_{\epsilon}({\mathcal G}))

be the composite of the maximal Baum-Connes map with the natural map from Ki(Cmax(G))K_i(C^*_{\max}({\mathcal G})) to Ki(Cϵ(G))K_i(C^*_{\epsilon}({\mathcal G})). Rectified Baum-Connes conjecture. The morphism μi,GBGW\mu^{BGW}_{i,{\mathcal G}} is an isomorphism for every iZ2i\in\mathbb Z_2. The conjecture is presented as the rectified replacement for the reduced Baum-Connes conjecture, whose groupoid version has counterexamples; the source states that Baum, Guentner and Willett established the relevant minimal crossed-product framework, but does not state that this conjecture is resolved in the generality above.

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Primary source

Moulay-Tahar Benameur and Victor Moulard, “Admissible Higson-Roe sequences for transformation groupoids”, arXiv:2411.00182 (2025).

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