Poincar\'e profiles of groups and spaces

TL;DR

This paper introduces Poincaré profiles as coarse invariants for metric measure spaces, linking them to growth, separation, and conformal dimension, and uses them to analyze embeddings.

Contribution

It defines Poincaré profiles, explores their properties for groups and hyperbolic spaces, and applies these invariants to study coarse embeddings.

Findings

Poincaré profiles determine space growth and separation.

Connections established between profiles and conformal dimension.

Applications show non-existence of certain coarse embeddings.

Abstract

We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincar\'{e} profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by Benjamini--Schramm--Tim\'{a}r. In this paper we focus on properties of the Poincar\'{e} profiles of groups with polynomial growth, and of hyperbolic spaces, where we deduce a connection between these profiles and conformal dimension. As applications, we use these invariants to show the non-existence of coarse embeddings in a variety of examples.