The twisted coarse Baum--Connes conjecture and relative hyperbolic groups

In this paper, we introduce a notion of stable coarse algebras for metric spaces with bounded geometry, and formulate the twisted coarse Baum--Connes conjecture with respect to stable coarse algebras. We prove permanence properties of this conjecture under coarse equivalences, unions and subspaces. As an application, we study higher index theory for a group G that is hyperbolic relative to a finite family of subgroups ₁, H₂, , HN\. We prove that G satisfies the twisted coarse Baum--Connes conjecture with respect to any stable coarse algebra if and only if each subgroup Hᵢ does.