Separation profile, isoperimetry, growth and compression
Annales de l'Institut Fourier, Online first, 49 p.

We give lower and upper bounds for the separation profile (introduced by Benjamini, Schramm & Timár) for various graphs using isoperimetric profile, volume growth and Hilbertian compression. For graphs which have polynomial isoperimetry and growth, we show that the separation profile is bounded between n a and n b for some a,b(0,1). For many amenable groups, we prove a lower bound of n/log(n) a for some a>1, and for groups admitting “good” embeddings into an  p space we prove an upper bound of n/log(n) b for some b(0,1). We show that solvable groups of exponential growth cannot have a separation profile bounded above by a sublinear power function. We also introduce a notion of local separation, with applications for percolation clusters of d and graphs which have polynomial isoperimetry and growth.

Pour différents types de graphes, nous établissons des bornes inférieures et supérieures sur leur profil de séparation (introduit par Benjamini, Schramm & Timár), en utilisant le profil isopérimétrique, la fonction de croissance et la compression dans les espaces de Hilbert. Dans le cas des graphes de dimension isopérimétrique supérieure à 1 et à croissance polynomiale, nous montrons que le profil de séparation est compris entre deux fonctions puissance, avec des exposants compris strictement entre 0 et 1. Pour de nombreux groupes moyennables, nous montrons une borne inférieure de la forme n/log(n) a avec a>1 et, pour les groupes ayant des «  bons  » plongements vers un espace  p une borne supérieure de la forme n/log(n) b avec b compris strictement entre 0 et 1. Nous prouvons que le profil de séparation d’un groupe résoluble à croissance exponentielle n’est jamais dominé par une fonction puissance sous-linéaire. Nous introduisons également une notion de séparation locale, avec des applications aux composantes de percolation de d et aux graphes de dimension isopééimétrique supérieure à 1 et àà croissance polynomiale.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3541
Classification: 0F65,  30L05,  05C25
Keywords: Cheeger constants, Coarse geometry, Geometric group theory
Gournay, Antoine 1; Le Coz, Corentin 2

1 Institut für Geometrie, TU Dresden, Zellescher Weg 12-14, 01069 Dresden, Germany
2 Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
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Gournay, Antoine; Le Coz, Corentin. Separation profile, isoperimetry, growth and compression. Annales de l'Institut Fourier, Online first, 49 p.

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