A relatively hyperbolic group satisfies the twisted coarse Baum-Connes conjecture with respect to any stable coarse algebra if and only if each peripheral subgroup does.
The N ovikov conjecture for groups with finite asymptotic dimension
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The classical Baum-Connes assembly map is quantitatively an isomorphism for lacunary hyperbolic groups containing large-girth graph sequences in their Cayley graphs.
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The twisted coarse Baum--Connes conjecture and relative hyperbolic groups
A relatively hyperbolic group satisfies the twisted coarse Baum-Connes conjecture with respect to any stable coarse algebra if and only if each peripheral subgroup does.
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Controlled Analytic Properties and the Quantitative Baum-Connes Conjecture
The classical Baum-Connes assembly map is quantitatively an isomorphism for lacunary hyperbolic groups containing large-girth graph sequences in their Cayley graphs.