JournalsrmiVol. 36, No. 6pp. 1835–1886

Poincaré profiles of groups and spaces

  • David Hume

    University of Oxford, UK
  • John M. Mackay

    University of Bristol, UK
  • Romain Tessera

    Université Paris Diderot, France
Poincaré profiles of groups and spaces cover

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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 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.

Cite this article

David Hume, John M. Mackay, Romain Tessera, Poincaré profiles of groups and spaces. Rev. Mat. Iberoam. 36 (2020), no. 6, pp. 1835–1886

DOI 10.4171/RMI/1184