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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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.