Separation profile, isoperimetry, growth and compression

Gournay, Antoine; Le Coz, Corentin

doi:10.5802/aif.3541

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

Annales de l'Institut Fourier, Online first, 49 p.

Abstract
Résumé

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 $a, b \in (0, 1)$ . 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 $ℓ^{p}$ space we prove an upper bound of $n / log {(n)}^{b}$ for some $b \in (0, 1)$ . 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 $ℤ^{d}$ 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 $ℓ^{p}$ 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 $ℤ^{d}$ et aux graphes de dimension isopééimétrique supérieure à $1$ et àà croissance polynomiale.

Received: 2020-01-02
Revised: 2021-10-25
Accepted: 2021-12-01
Online First: 2022-11-28

DOI: 10.5802/aif.3541

Classification: 0F65, 30L05, 05C25
Keywords: Cheeger constants, Coarse geometry, Geometric group theory

Author's affiliations:

Gournay, Antoine ¹; Le Coz, Corentin ²

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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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