Coarse separation and the geometry of lamplighter groups

A subset S of a metric space X is said to coarsely separate X if the complement of a D-neighborhood of S contains at least two connected components with arbitrarily large balls. We are interested in the following question: Which spaces of exponential volume growth do not admit a separating subset of sub-exponential volume growth? We start by describing how this notion of separation arises naturally in the quasi-isometry classification of wreath products, also called lamplighter groups. Then we show that symmetric spaces of non-compact type (except the real hyperbolic plane), and some buildings and horocyclic products do not admit a separating subset of sub-exponential growth. Joint work with Anthony Genevois and Romain Tessera.