In a joint work with Bensaid and Genevois, we characterize among a vast class of groups those which can be coarsely separated by subsets of subexponential growth. For instance, we prove that such a hyperbolic group must virtually split over a cyclic group, or that the defining graph of such a RAAG must be separated by a clique. Our notion of coarse separation arises from the study of coarse embeddings into wreath products H\wr G with lamp group H of subexponential growth or into amalgamated products over non-distorted subgroups of subexponential growth.
