Title: Poincaré profiles of groups and spaces,
Abstract: We introduce a spectrum of monotone coarse invariants for metric measure
spaces called Poincaré profiles. The two extremes of this spectrum
determine the growth of the space, and the separation profile as defined by
Benjamini--Schramm--Timár. In this paper we focus on properties of the
Poincaré profiles of groups with polynomial growth, and of hyperbolic
spaces, where we deduce a striking connection between these profiles and
conformal dimension. One application of our results is that there is a
collection of hyperbolic Coxeter groups, indexed by a countable dense subset of
$(1,\infty)$, such that $G_s$ does not coarsely embed into $G_t$ whenever
$s<t$.