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.