The equivariant coarse Baum-Connes conjecture for (X, Γ) is equivalent to the groupoid Baum-Connes conjecture for the associated equivariant coarse groupoid G(X, Γ), implying the equivariant coarse Novikov conjecture when X coarsely embeds into Hilbert space.
Almost elementariness and fiberwise amenability for ´ etale groupoids.arXiv: 2011.01182
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Minimal ample groupoids exist that are neither almost finite nor purely infinite, including amenable transformation groupoids and new essentially principal ones built from twisted topological groupoid constructions.
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A groupoid approach to the equivariant coarse Baum--Connes conjecture
The equivariant coarse Baum-Connes conjecture for (X, Γ) is equivalent to the groupoid Baum-Connes conjecture for the associated equivariant coarse groupoid G(X, Γ), implying the equivariant coarse Novikov conjecture when X coarsely embeds into Hilbert space.
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Ample groupoids that are neither almost finite nor purely infinite
Minimal ample groupoids exist that are neither almost finite nor purely infinite, including amenable transformation groupoids and new essentially principal ones built from twisted topological groupoid constructions.