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 and for some . For many amenable groups, we prove a lower bound of for some , and for groups admitting “good” embeddings into an space we prove an upper bound of for some . 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 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 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 avec et, pour les groupes ayant des « bons » plongements vers un espace une borne supérieure de la forme avec 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 et aux graphes de dimension isopééimétrique supérieure à et àà croissance polynomiale.
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Keywords: Cheeger constants, Coarse geometry, Geometric group theory
@unpublished{AIF_0__0_0_A131_0, author = {Gournay, Antoine and Le Coz, Corentin}, title = {Separation profile, isoperimetry, growth and compression}, journal = {Annales de l'Institut Fourier}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, year = {2022}, doi = {10.5802/aif.3541}, language = {en}, note = {Online first}, }
TY - UNPB AU - Gournay, Antoine AU - Le Coz, Corentin TI - Separation profile, isoperimetry, growth and compression JO - Annales de l'Institut Fourier PY - 2022 DA - 2022/// PB - Association des Annales de l’institut Fourier N1 - Online first UR - https://doi.org/10.5802/aif.3541 DO - 10.5802/aif.3541 LA - en ID - AIF_0__0_0_A131_0 ER -
%0 Unpublished Work %A Gournay, Antoine %A Le Coz, Corentin %T Separation profile, isoperimetry, growth and compression %J Annales de l'Institut Fourier %D 2022 %I Association des Annales de l’institut Fourier %Z Online first %U https://doi.org/10.5802/aif.3541 %R 10.5802/aif.3541 %G en %F AIF_0__0_0_A131_0
Gournay, Antoine; Le Coz, Corentin. Separation profile, isoperimetry, growth and compression. Annales de l'Institut Fourier, Online first, 49 p.
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