Poincar\'e profiles of Lie groups and a coarse geometric dichotomy

Poincar\'e profiles are a family of analytically defined coarse invariants, which can be used as obstructions to the existence of coarse embeddings between metric spaces. In this paper we calculate the Poincar\'e profiles of all connected unimodular Lie groups, Baumslag-Solitar groups and Thurston geometries, demonstrating two substantially different types of behaviour. In the case of Lie groups, we obtain a dichotomy which extends both the dichotomy separating rank one and higher rank semisimple Lie groups and the dichotomy separating connected solvable unimodular Lie groups of polynomial and exponential growth. We provide equivalent algebraic, quasi-isometric and coarse geometric formulations of this dichotomy. Our results have many consequences for coarse embeddings, for instance we deduce that for groups of the form $N\times S$, where $N$ is a connected nilpotent Lie group, and $S$ is a simple Lie group of real rank 1, both the growth exponent of $N$, and the Ahlfors-regular conformal dimension of $S$ are non-decreasing under coarse embeddings. These results are new even in the quasi-isometric setting and give obstructions to quasi-isometric embeddings which in many cases are stronger than those previously obtained by Buyalo-Schroeder.