A groupoid approach to the equivariant coarse Baum--Connes conjecture

2026/04/30 by Liang Guo
Mathematics · #math.KT #math.MG #math.OA #msc:58J22 #msc:22A22 #msc:51F30 #msc:19K35

paper · pdf

35 pages

arxiv created 2026/07/31 · arxiv updated 2026/08/03

Abstract

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 ℓ^∞(X,K)^Γ is equivalent to the equivariant coarse Baum--Connes conjecture for (X, Γ) using a localization algebra description of equivariant KK^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.

Citations

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