Benjamini, Schramm and Timár quantified how well-connected an infinite graph is in terms of its "separation profile", where one considers the cut size of finite subgraphs. Instead of cut size one can use the spectral gap of the graph p-Laplacian to measure the connectivity of finite subgraphs, and these "p-Poincaré profiles" were used in previous work with Hume and Tessera to show a variety of non-embedding results between groups. I'll mainly talk about current work with Hume where we further study the connection between these profiles and the conformal dimension of the boundary at infinity of certain Gromov hyperbolic groups.
