Poincaré profiles of groups and spaces

David Hume; John M. Mackay; Romain Tessera

doi:10.4171/rmi/1184

Poincaré profiles of groups and spaces

David Hume, John M. Mackay, Romain Tessera

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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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.

Original language	English
Pages (from-to)	1835-1886
Number of pages	52
Journal	Revista Matemática Iberoamericana
Volume	36
Issue number	6
Early online date	2 Mar 2020
DOIs	https://doi.org/10.4171/rmi/1184
Publication status	E-pub ahead of print - 2 Mar 2020

Keywords

hyperbolic groups
nilpotent groups
Poincaré inequality
expanders
coarse invariant

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10.4171/rmi/1184

Full-text PDF (accepted author manuscript)
This is the author accepted manuscript (AAM). The final published version (version of record) is available online via European Mathematical Society at https://www.ems-ph.org/journals/of_article.php?jrn=RMI&doi=1184 . Please refer to any applicable terms of use of the publisher.
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https://arxiv.org/abs/1707.02151#:~:text=Poincar%C3%A9%20profiles%20of%20groups%20and%20spaces,-David%20Hume%2C%20John&text=We%20introduce%20a%20spectrum%20of,Benjamini%2D%2DSchramm%2D%2DTim%C3%A1r.

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Optimal geometric structures for hyperbolic groups
Mackay, J. M. (Principal Investigator)
1/11/16 → 31/08/18
Project: Research

Dr John M Mackay
- john.mackaybristol.acuk
- School of Mathematics - Associate Professor in Pure Mathematics
- Probability, Analysis and Dynamics
- Pure Mathematics
- Analysis
Person: Academic , Member

Cite this

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title = "Poincar{\'e} profiles of groups and spaces",

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 = "hyperbolic groups, nilpotent groups, Poincar{\'e} inequality, expanders, coarse invariant",

author = "David Hume and Mackay, {John M.} and Romain Tessera",

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doi = "10.4171/rmi/1184",

language = "English",

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T1 - Poincaré profiles of groups and spaces

AU - Hume, David

AU - Mackay, John M.

AU - Tessera, Romain

PY - 2020/3/2

Y1 - 2020/3/2

N2 - 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.

AB - 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.

KW - hyperbolic groups

KW - nilpotent groups

KW - Poincaré inequality

KW - expanders

KW - coarse invariant

U2 - 10.4171/rmi/1184

DO - 10.4171/rmi/1184

M3 - Article (Academic Journal)

SN - 0213-2230

VL - 36

SP - 1835

EP - 1886

JO - Revista Matemática Iberoamericana

JF - Revista Matemática Iberoamericana

IS - 6

ER -

Poincaré profiles of groups and spaces

Abstract

Keywords

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Optimal geometric structures for hyperbolic groups

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Dr John M Mackay

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