Résume | A subset S of a metric space X is coarsely separating if there exists a constant D such that for any R, at least two connected components of the complement of the D-neighborhood of S contain balls of radius R. We are interested in the following questions: Can we coarsely separate a connected nilpotent Lie group of growth degree d by a subspace of growth degree d-2? Which spaces of exponential growth do not admit a separating subset of subexponential growth? We start by describing how this separation property arises naturally when dealing with the large-scale geometry of amalgamated free products and wreath products. Then we answer the first question in the negative, and give examples of spaces answering the second question, namely all symmetric spaces of non-compact type (except the real hyperbolic plane), higher rank thick Euclidean buildings, and some SOL type geometries. This is a joint work with Anthony Genevois and Romain Tessera. |