  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.
