Abstract: The separation profile of an infinite graph was introduced by Benjamini-Schramm-Timar. It is a function which measures how well-connected the graph is by how hard it is to cut finite subgraphs into small pieces. In earlier joint work with David Hume and Romain Tessera, we introduced Poincaré profiles, generalising this concept by using p-Poincaré inequalities to measure the connected-ness of subgraphs. I will discuss this family of invariants, their applications to coarse embedding problems, and recent work finding the profiles of all connected unimodular Lie groups, where a dichotomy is exhibited.

Joint with Hume and Tessera.