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 $\{H_1, H_2, \dots, H_N\}$. We prove that $G$ satisfies the twisted coarse Baum--Connes conjecture with respect to any stable coarse algebra if and only if each subgroup $H_i$ does.