no. 4
Isoperimetric inequalities, shapes of Følner sets and groups with Shalom’s property H FD
[Les inégalités isopérimétriques, la forme des ensembles de Følner et des groupes avec la propriété H FD de Shalom.]
Erschler, Anna1 ; Zheng, Tianyi2
1 Département de mathématiques et applications, École normale supérieure, CNRS PSL Research University 45 rue d’Ulm 75005 Paris (France)
2 Department of Mathematics UC San Diego 9500 Giman Dr. La Jolla CA 92093 (USA)
Annales de l'Institut Fourier, Tome 70 (2020) no. 4, pp. 1363-1402.

Nous prouvons une inégalité isopérimétrique pour les groupes. En application, nous montrons que la fonction de Følner de tout groupe de Grigorchuk à croissance intermédiaire est au moins exponentielle. En tant qu’autre application, nous obtenons des bornes inférieures sur les fonctions de Følner dans divers groupes nilpotents par cycliques. Sous une hypothèse de régularité, nous obtenons une caractérisation des fonctions de Følner de ces groupes. Comme autre application, nous évaluons le comportement asymptotique de la fonction de Følner de Sym(ℤ)⋊ℤ. Nous étudions des exemples de groupes avec la propriété de Shalom H FD parmi les extensions d’un groupe nilpotent par un groupe cyclique. Nous montrons qu’il existe des groupes hyperboliques lacunaires avec la propriété H FD . Nous trouvons des groupes avec la propriété H FD , qui sont des produits directs de groupes hyperbolique lacunaires et ont des fonctions Følner arbitrairement grandes.

We prove an isoperimetric inequality for groups. As an application we show that any Grigrochuk group of intermediate growth has at least exponential Følner function. As another application, we obtain lower bounds on Følner functions in various nilpotent-by-cyclic groups. Under a regularity assumption, we obtain a characterization of Følner functions of these groups. As a further application, we evaluate the asymptotics of the Følner function of Sym(ℤ)⋊ℤ. We study examples of groups with Shalom’s property H FD among nilpotent-by-cyclic groups. We show that there exist lacunary hyperbolic groups with property H FD . We find groups with property H FD , which are direct products of lacunary hyperbolic groups and have arbitrarily large Følner functions.

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DOI : 10.5802/aif.3360
Classification : 20F65, 20E22, 60B15
Keywords: Følner function, isoperimetric profile, Følner sets, growth function
Mot clés : Fonction de Folner, profil isopérimétrique, ensembles de Folner, fonction de croissance

Erschler, Anna 1 ; Zheng, Tianyi 2

1 Département de mathématiques et applications, École normale supérieure, CNRS PSL Research University 45 rue d’Ulm 75005 Paris (France)
2 Department of Mathematics UC San Diego 9500 Giman Dr. La Jolla CA 92093 (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     journal = {Annales de l'Institut Fourier},
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Erschler, Anna; Zheng, Tianyi. Isoperimetric inequalities, shapes of Følner sets and groups with Shalom’s property ${H_{\protect \mathrm{FD}}}$. Annales de l'Institut Fourier, Tome 70 (2020) no. 4, pp. 1363-1402. doi : 10.5802/aif.3360. https://aif.centre-mersenne.org/articles/10.5802/aif.3360/

[1] Amir, Gideon; Virág, Bálint Speed Exponents of Random Walks on Groups, Int. Math. Res. Not. (2017) no. 9, pp. 2567-2598 | MR | Zbl

[2] Bartholdi, Laurent; Grigorčuk, Rostislav I.; Šuniḱ, Zoran Branch groups, Handbook of algebra, Vol. 3, Volume 3, Elsevier, 2003, pp. 989-1112 | DOI | MR | Zbl

[3] Brieussel, Jérémie; Zheng, Tianyi Speed of random walks, isoperimetry and compression of finitely generated groups (2015) (https://arxiv.org/abs/1510.08040) | Zbl

[4] Brieussel, Jérémie; Zheng, Tianyi Shalom’s property H FD and extensions by ℤ by locally finite groups (2017) (https://arxiv.org/abs/1706.00707) | Zbl

[5] Coulhon, Thierry; Grigor’yan, Alexander; Pittet, Christophe A geometric approach to on-diagonal heat kernel lower bounds on groups, Ann. Inst. Fourier, Volume 51 (2001) no. 6, pp. 1763-1827 | DOI | MR | Zbl

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

[7] Coulon, Rémi; Guirardel, Vincent Automorphisms and endomorphisms of lacunary hyperbolic groups (2016) (https://arxiv.org/abs/1606.00679) | Zbl

[8] Delorme, Patrick 1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations, Bull. Soc. Math. Fr., Volume 105 (1977) no. 3, pp. 281-336 | DOI | Numdam | Zbl

[9] Deroin, Bertrand; Navas, Andrés; Rivas, Cristóbal Groups, orders, and dynamics (2014) (https://arxiv.org/abs/1408.5805)

[10] van den Dries, Lou; Wilkie, Alex J. Gromov’s theorem on groups of polynomial growth and elementary logic, J. Algebra, Volume 89 (1984) no. 2, pp. 349-374 | DOI | MR | Zbl

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

[12] Erschler, Anna Critical constants for recurrence of random walks on G-spaces, Ann. Inst. Fourier, Volume 55 (2005) no. 2, pp. 493-509 | DOI | Numdam | MR | Zbl

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

[14] Erschler, Anna Poisson–Furstenberg boundaries, large-scale geometry and growth of groups, Proceedings of the International Congress of Mathematicians. Volume II (2010), pp. 681-704 | MR | Zbl

[15] Erschler, Anna Almost invariance of distributions for random walks on groups (2016) (https://arxiv.org/abs/1603.01458) | Zbl

[16] Erschler, Anna; Ozawa, Narutaka Finite-Dimensional Representations constructed from Random Walks (2016) (https://arxiv.org/abs/1609.08585) | Zbl

[17] Gournay, Antoine Mixing, malnormal subgroups and cohomology in degree one (2016) (https://arxiv.org/abs/1607.05056) | Zbl

[18] Grigorčuk, Rostislav I. On Burnside’s problem on periodic groups, Funkts. Anal. Prilozh., Volume 14 (1980) no. 1, pp. 53-54 | MR | Zbl

[19] Grigorčuk, Rostislav I. Degrees of growth of finitely generated groups, and the theory of invariant means, Izv. Math., Volume 25 (1985) no. 2, pp. 259-300 | DOI | Zbl

[20] Grigorčuk, Rostislav I. On growth in group theory, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) (1991), pp. 325-338 | MR

[21] Grigorčuk, Rostislav I. Milnor’s problem on the growth of groups and its consequences, Frontiers in complex dynamics (Princeton Mathematical Series), Volume 51, Princeton University Press, 2014, pp. 705-773 | DOI | MR

[22] Gromov, Misha Groups of polynomial growth and expanding maps, Publ. Math., Inst. Hautes Étud. Sci. (1981) no. 53, pp. 53-73 | DOI | Numdam | MR

[23] Gromov, Misha Hyperbolic groups, Mathematical Sciences Research Institute Publications, 8, Springer, 1987, pp. 75-263 | MR

[24] Gromov, Misha Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, 152, Birkhäuser, 1999, xx+585 pages | MR | Zbl

[25] 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

[26] Hall, Marshall Jr. The theory of groups, The Macmillan Co., 1959, xiii+434 pages | Zbl

[27] Hall, Philip Finiteness conditions for soluble groups, Proc. Lond. Math. Soc., Volume 4 (1954), pp. 419-436 | DOI | MR | Zbl

[28] de La Harpe, Pierre Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, 2000 | Zbl

[29] Ol’shanskii, Alexander Yu.; Osin, Denis V. A quasi-isometric embedding theorem for groups, Duke Math. J., Volume 162 (2013) no. 9, pp. 1621-1648 | MR | Zbl

[30] Ol’shanskii, Alexander Yu.; Osin, Denis V.; Sapir, Mark V. Lacunary hyperbolic groups, Geom. Topol., Volume 13 (2009) no. 4, pp. 2051-2140 (with an appendix by Michael Kapovich and Bruce Kleiner) | DOI | MR | Zbl

[31] Osin, Denis V. Kazhdan constants of hyperbolic groups, Funct. Anal. Appl., Volume 36 (2002) no. 4, pp. 290-297 | DOI | MR | Zbl

[32] Ozawa, Narutaka A functional analysis proof of Gromov’s polynomial growth theorem, Ann. Sci. Éc. Norm. Supér., Volume 51 (2018) no. 3, pp. 549-556 | DOI | MR | Zbl

[33] Saloff-Coste, Laurent; Zheng, Tianyi Isoperimetric profiles and random walks on some permutation wreath products (2015) (https://arxiv.org/abs/1510.08830) | Zbl

[34] Serre, Jean-Pierre Trees, Springer Monographs in Mathematics, Springer, 2003, x+142 pages (translated from the French original by John Stillwell, corrected 2nd printing of the 1980 English translation) | Zbl

[35] Shalom, Yehuda Harmonic analysis, cohomology, and the large-scale geometry of amenable groups, Acta Math., Volume 192 (2004) no. 2, pp. 119-185 | DOI | MR | Zbl

[36] Tessera, Romain Large scale Sobolev inequalities on metric measure spaces and applications, Rev. Mat. Iberoam., Volume 24 (2008) no. 3, pp. 825-864 | DOI | MR | Zbl

[37] 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

[38] 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

[39] Woess, Wolfgang Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, 138, Cambridge University Press, 2000, xii+334 pages | MR | Zbl

[40] Yadin, Ariel Rate of escape of the mixer chain, Electron. Commun. Probab., Volume 14 (2009), pp. 347-357 | MR | Zbl

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