Poincar\'e profiles of groups and spaces
Group Theory
2020-11-09 v3 Geometric Topology
Metric Geometry
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.
Keywords
Cite
@article{arxiv.1707.02151,
title = {Poincar\'e profiles of groups and spaces},
author = {David Hume and John M. Mackay and Romain Tessera},
journal= {arXiv preprint arXiv:1707.02151},
year = {2020}
}
Comments
55 pages. To appear in Revista Matem\'atica Iberoamericana