Separation profile, isoperimetry, growth and compression

Gournay, Antoine; Le Coz, Corentin

doi:10.5802/aif.3541

Separation profile, isoperimetry, growth and compression
[Profil de séparation, isopérimétrie, croissance et compression]

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, Tome 73 (2023) no. 4, pp. 1627-1675.

Abstract (VO)
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 isopérimétrique supérieure à $1$ et à croissance polynomiale.

Reçu le : 2020-01-02
Révisé le : 2021-10-25
Accepté le : 2021-12-01
Publié le : 2023-07-07

DOI : 10.5802/aif.3541

Classification : 20F65, 30L05, 05C25
Keywords: Cheeger constants, Coarse geometry, Geometric group theory
Mots-clés : Constantes de Cheeger, géométrie grossière, théorie géométrique des groupes.

Affiliations des auteurs :

Gournay, Antoine ¹ ; Le Coz, Corentin ²

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Separation profile, isoperimetry, growth and compression},
     journal = {Annales de l'Institut Fourier},
     pages = {1627--1675},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {73},
     number = {4},
     year = {2023},
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Gournay, Antoine; Le Coz, Corentin. Separation profile, isoperimetry, growth and compression. Annales de l'Institut Fourier, Tome 73 (2023) no. 4, pp. 1627-1675. doi : 10.5802/aif.3541. https://aif.centre-mersenne.org/articles/10.5802/aif.3541/

Bibliographie
Cité par

[1] Amghibech, S. Eigenvalues of the discrete $p$ -Laplacian for graphs, Ars Combin., Volume 67 (2003), pp. 283-302 | MR | Zbl

[2] Arzhantseva, G. N.; Guba, V. S.; Sapir, M. V. Metrics on diagram groups and uniform embeddings in a Hilbert space, Comment. Math. Helv., Volume 81 (2006) no. 4, pp. 911-929 | DOI | MR | Zbl

[3] Barlow, Martin T. Random walks on supercritical percolation clusters, Ann. Probab., Volume 32 (2004) no. 4, pp. 3024-3084 | DOI | MR | Zbl

[4] Bendikov, Alexander; Pittet, Christophe; Sauer, Roman Spectral distribution and $L^{2}$ -isoperimetric profile of Laplace operators on groups, Math. Ann., Volume 354 (2012) no. 1, pp. 43-72 | DOI | MR | Zbl

[5] Benjamini, Itai; Mossel, Elchanan On the mixing time of a simple random walk on the super critical percolation cluster, Probab. Theory Related Fields, Volume 125 (2003) no. 3, pp. 408-420 | DOI | MR | Zbl

[6] Benjamini, Itai; Papasoglu, Panos Growth and isoperimetric profile of planar graphs, Proc. Amer. Math. Soc., Volume 139 (2011) no. 11, pp. 4105-4111 | DOI | MR | Zbl

[7] Benjamini, Itai; Schramm, Oded; Timár, Ádám On the separation profile of infinite graphs, Groups Geom. Dyn., Volume 6 (2012) no. 4, pp. 639-658 | DOI | MR | Zbl

[8] Bonk, Mario; Schramm, Oded Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal., Volume 10 (2000) no. 2, pp. 266-306 | DOI | MR | Zbl

[9] Brieussel, Jérémie; Zheng, Tianyi Speed of random walks, isoperimetry and compression of finitely generated groups, Ann. of Math. (2), Volume 193 (2021) no. 1, pp. 1-105 | DOI | MR | Zbl

[10] Buyalo, Sergei; Dranishnikov, Alexander; Schroeder, Viktor Embedding of hyperbolic groups into products of binary trees, Invent. Math., Volume 169 (2007) no. 1, pp. 153-192 | DOI | MR | Zbl

[11] Buyalo, Sergei; Schroeder, Viktor Elements of asymptotic geometry, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007, xii+200 pages | DOI | MR | Zbl

[12] Coulhon, Thierry; Saloff-Coste, Laurent Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamericana, Volume 9 (1993) no. 2, pp. 293-314 | DOI | MR | Zbl

[13] Erschler, Anna On isoperimetric profiles of finitely generated groups, Geom. Dedicata, Volume 100 (2003), pp. 157-171 | MR | Zbl

[14] Erschler, Anna Piecewise automatic groups, Duke Math. J., Volume 134 (2006) no. 3, pp. 591-613 | DOI | MR | Zbl

[15] Gentimis, Thanos Asymptotic dimension of finitely presented groups, Proc. Amer. Math. Soc., Volume 136 (2008) no. 12, pp. 4103-4110 | DOI | MR | Zbl

[16] Gibson, Lee R.; Pivarski, Melanie Isoperimetric profiles on the pre-fractal Sierpinski carpet, Fractals, Volume 18 (2010) no. 4, pp. 433-449 | DOI | MR | Zbl

[17] Gladkova, Valeriia; Shum, Verna Separation profiles of graphs of fractals (2018) (https://arxiv.org/abs/1810.08792)

[18] Gournay, Antoine The Liouville property and Hilbertian compression, Ann. Inst. Fourier (Grenoble), Volume 66 (2016) no. 6, pp. 2435-2454 | DOI | MR | Zbl

[19] Gromov, Misha Entropy and isoperimetry for linear and non-linear group actions, Groups Geom. Dyn., Volume 2 (2008) no. 4, pp. 499-593 | DOI | MR | Zbl

[20] de la Harpe, Pierre Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000, vi+310 pages | MR | Zbl

[21] Hume, David Direct embeddings of relatively hyperbolic groups with optimal $ℓ^{p}$ compression exponent, J. Reine Angew. Math., Volume 703 (2015), pp. 147-172 | DOI | MR | Zbl

[22] Hume, David A continuum of expanders, Fund. Math., Volume 238 (2017) no. 2, pp. 143-152 | DOI | MR | Zbl

[23] Hume, David; Mackay, John M. Poorly connected groups, Proc. Amer. Math. Soc., Volume 148 (2020) no. 11, pp. 4653-4664 | DOI | MR | Zbl

[24] Hume, David; Mackay, John M.; Tessera, Romain Poincaré profiles of groups and spaces, Rev. Mat. Iberoam., Volume 36 (2020) no. 6, pp. 1835-1886 | DOI | MR | Zbl

[25] Hume, David; Sisto, Alessandro Groups with no coarse embeddings into hyperbolic groups, New York J. Math., Volume 23 (2017), pp. 1657-1670 | MR | Zbl

[26] Jolissaint, Pierre-Nicolas; Pillon, Thibault $L^{p}$ compression of some HNN extensions, J. Group Theory, Volume 16 (2013) no. 6, pp. 907-913 | DOI | MR | Zbl

[27] Jolissaint, Pierre-Nicolas; Valette, Alain $L^{p}$ -distortion and $p$ -spectral gap of finite graphs, Bull. Lond. Math. Soc., Volume 46 (2014) no. 2, pp. 329-341 | DOI | MR | Zbl

[28] Li, Sean Compression bounds for wreath products, Proc. Amer. Math. Soc., Volume 138 (2010) no. 8, pp. 2701-2714 | DOI | MR | Zbl

[29] Loomis, L. H.; Whitney, H. An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc, Volume 55 (1949), pp. 961-962 | DOI | MR | Zbl

[30] Naor, Assaf; Peres, Yuval Embeddings of discrete groups and the speed of random walks, Int. Math. Res. Not. IMRN (2008), 076, 34 pages | DOI | MR | Zbl

[31] Naor, Assaf; Peres, Yuval $L_{p}$ compression, traveling salesmen, and stable walks, Duke Math. J., Volume 157 (2011) no. 1, pp. 53-108 | DOI | MR | Zbl

[32] Pete, Gábor A note on percolation on $ℤ^{d}$ : isoperimetric profile via exponential cluster repulsion, Electron. Commun. Probab., Volume 13 (2008), pp. 377-392 | DOI | MR | Zbl

[33] Pittet, Christophe; Saloff-Coste, L. Random walks on finite rank solvable groups, J. Eur. Math. Soc. (JEMS), Volume 5 (2003) no. 4, pp. 313-342 | DOI | MR | Zbl

[34] Pittet, Christophe; Saloff-Coste, Laurent Amenable groups, isoperimetric profiles and random walks, Geometric group theory down under (Canberra, 1996), de Gruyter, Berlin, 1999, pp. 293-316 | MR | Zbl

[35] Pittet, Christophe; Saloff-Coste, Laurent A survey on the relationships between volume growth, isoperimetry, and the behavior of simple random walk on Cayley graphs, with examples (2001) (https://math.unice.fr/~indira/papers/surveyPS.pdf)

[36] Sale, Andrew W. Metric behaviour of the Magnus embedding, Geom. Dedicata, Volume 176 (2015), pp. 305-313 | DOI | MR | Zbl

[37] Saloff-Coste, Laurent; Zheng, Tianyi Random walks and isoperimetric profiles under moment conditions, Ann. Probab., Volume 44 (2016) no. 6, pp. 4133-4183 | DOI | MR | Zbl

[38] Saloff-Coste, Laurent; Zheng, Tianyi Isoperimetric profiles and random walks on some permutation wreath products, Rev. Mat. Iberoam., Volume 34 (2018) no. 2, pp. 481-540 | DOI | MR | Zbl

[39] Tessera, Romain Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces, Comment. Math. Helv., Volume 86 (2011) no. 3, pp. 499-535 | DOI | MR | Zbl

[40] Tessera, Romain Isoperimetric profile and random walks on locally compact solvable groups, Rev. Mat. Iberoam., Volume 29 (2013) no. 2, pp. 715-737 | DOI | MR | Zbl

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