Super-expanders and warped cones
Annales de l'Institut Fourier, Volume 70 (2020) no. 4, pp. 1753-1774.

For a Banach space X, we show that any family of graphs quasi-isometric to levels of a warped cone 𝒪 Γ Y is an expander with respect to X if and only if the induced Γ-representation on L 2 (Y;X) has a spectral gap. This provides examples of graphs that are an expander with respect to all Banach spaces of non-trivial type.

Pour un espace de Banach X, on montre que toute famille de graphes, quasi-isométriques à des niveaux d’un cône tordu 𝒪 Γ Y, est un expanseur relativement à X, si et seulement si la Γ-représentation induite sur L 2 (Y;X) a un trou spectral. Ceci fournit des examples de graphes qui sont un expanseur relativement à tous les espaces de Banach de type non trivial.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3373
Classification: 46B85,  05C25,  37A30,  37C85
Keywords: expander graph, spectral gap, warped cone, quasi-isometry, Rademacher type.
Sawicki, Damian 1

1 Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-656 Warszawa (Poland)
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Sawicki, Damian. Super-expanders and warped cones. Annales de l'Institut Fourier, Volume 70 (2020) no. 4, pp. 1753-1774. doi : 10.5802/aif.3373. https://aif.centre-mersenne.org/articles/10.5802/aif.3373/

[1] Benoist, Yves; de Saxcé, Nicolas A spectral gap theorem in simple Lie groups, Invent. Math., Volume 205 (2016) no. 2, pp. 337-361 | Article | MR: 3529116

[2] Bourgain, Jean Expanders and dimensional expansion, C. R. Math. Acad. Sci. Paris, Volume 347 (2009) no. 7-8, pp. 357-362 | Article | MR: 2537230

[3] Bourgain, Jean; Gamburd, Alex On the spectral gap for finitely-generated subgroups of SU (2), Invent. Math., Volume 171 (2008) no. 1, pp. 83-121 | Article | MR: 2358056

[4] Bourgain, Jean; Gamburd, Alex Uniform expansion bounds for Cayley graphs of SL 2 (𝔽 p ), Ann. Math., Volume 167 (2008) no. 2, pp. 625-642 | Article | MR: 2415383

[5] Bourgain, Jean; Gamburd, Alex A spectral gap theorem in SU(d), J. Eur. Math. Soc., Volume 14 (2012) no. 5, pp. 1455-1511 | Article | MR: 2966656

[6] Bourgain, Jean; Yehudayoff, Amir Expansion in SL 2 () and monotone expanders, Geom. Funct. Anal., Volume 23 (2013) no. 1, pp. 1-41 | Article | MR: 3037896

[7] Boutonnet, Rémi; Ioana, Adrian; Golsefidy, Alireza Salehi Local spectral gap in simple Lie groups and applications, Invent. Math., Volume 208 (2017) no. 3, pp. 715-802 | Article | MR: 3648974

[8] Charles, Denis X.; Lauter, Kristin E.; Goren, Eyal Z. Cryptographic hash functions from expander graphs, J. Cryptology, Volume 22 (2009) no. 1, pp. 93-113 | Article | MR: 2496385 | Zbl: 1166.94006

[9] Cheng, Qingjin Sphere equivalence, property H, and Banach expanders, Stud. Math., Volume 233 (2016) no. 1, pp. 67-83 | MR: 3503071

[10] Diestel, Joe; Jarchow, Hans; Tonge, Andrew Absolutely summing operators, Cambridge Studies in Advanced Mathematics, 43, Cambridge University Press, 1995, xvi+474 pages | Article | MR: 1342297

[11] Druţu, Cornelia; Nowak, Piotr W. Kazhdan projections, random walks and ergodic theorems, J. Reine Angew. Math., Volume 754 (2019), pp. 49-86 | Article | MR: 4000570 | Zbl: 1429.46036

[12] Fisher, David; Nguyen, Thang; van Limbeek, Wouter Rigidity of warped cones and coarse geometry of expanders, Adv. Math., Volume 346 (2019), pp. 665-718 | Article | MR: 3912479 | Zbl: 1440.53047

[13] Gabber, Ofer; Galil, Zvi Explicit constructions of linear-sized superconcentrators, J. Comput. Syst. Sci., Volume 22 (1981) no. 3, pp. 407-420 (Special issued dedicated to Michael Machtey) | Article | MR: 633542 | Zbl: 0487.05045

[14] Higson, Nigel; Lafforgue, Vincent; Skandalis, Georges Counterexamples to the Baum–Connes conjecture, Geom. Funct. Anal., Volume 12 (2002) no. 2, pp. 330-354 | Article | MR: 1911663 | Zbl: 1014.46043

[15] Hoory, Shlomo; Linial, Nathan; Wigderson, Avi Expander graphs and their applications, Bull. Am. Math. Soc., Volume 43 (2006) no. 4, pp. 439-561 | Article | MR: 2247919 | Zbl: 1147.68608

[16] Kasparov, Gennadi; Yu, Guoliang The coarse geometric Novikov conjecture and uniform convexity, Adv. Math., Volume 206 (2006) no. 1, pp. 1-56 | Article | MR: 2261750 | Zbl: 1102.19003

[17] Khukhro, Ana; Valette, Alain Expanders and box spaces, Adv. Math., Volume 314 (2017), pp. 806-834 | Article | MR: 3658731 | Zbl: 1426.20015

[18] de Laat, Tim; de la Salle, Mikael Strong property (T) for higher-rank simple Lie groups, Proc. Lond. Math. Soc., Volume 111 (2015) no. 4, pp. 936-966 | Article | MR: 3407190 | Zbl: 1328.22003

[19] de Laat, Tim; Vigolo, Federico Superexpanders from group actions on compact manifolds, Geom. Dedicata, Volume 200 (2019), pp. 287-302 | Article | MR: 3956196 | Zbl: 07065313

[20] Lafforgue, Vincent Un renforcement de la propriété (T), Duke Math. J., Volume 143 (2008) no. 3, pp. 559-602 | Article | MR: 2423763 | Zbl: 1158.46049

[21] Lafforgue, Vincent Propriété (T) renforcée banachique et transformation de Fourier rapide, J. Topol. Anal., Volume 1 (2009) no. 3, pp. 191-206 | Article | MR: 2574023 | Zbl: 1186.46022

[22] Liao, Benben Strong Banach property (T) for simple algebraic groups of higher rank, J. Topol. Anal., Volume 6 (2014) no. 1, pp. 75-105 | Article | MR: 3190138 | Zbl: 1291.22010

[23] Lubotzky, Alexander Expander graphs in pure and applied mathematics, Bull. Am. Math. Soc., Volume 49 (2012) no. 1, pp. 113-162 | Article | MR: 2869010 | Zbl: 1232.05194

[24] Lubotzky, Alexander; Phillips, Ralph; Sarnak, Peter Ramanujan graphs, Combinatorica, Volume 8 (1988) no. 3, pp. 261-277 | Article | MR: 963118 | Zbl: 0661.05035

[25] Margulis, Grigorii Aleksandrovich Explicit constructions of expanders, Probl. Peredachi Inf., Volume 9 (1973) no. 4, pp. 71-80 | MR: 484767

[26] Margulis, Grigorii Aleksandrovich Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators, Probl. Peredachi Inf., Volume 24 (1988) no. 1, pp. 51-60 | MR: 939574

[27] Mendel, Manor; Naor, Assaf Nonlinear spectral calculus and super-expanders, Publ. Math., Inst. Hautes Étud. Sci., Volume 119 (2014), pp. 1-95 | Article | Numdam | MR: 3210176 | Zbl: 1306.46021

[28] Mendel, Manor; Naor, Assaf Expanders with respect to Hadamard spaces and random graphs, Duke Math. J., Volume 164 (2015) no. 8, pp. 1471-1548 | Article | MR: 3352040 | Zbl: 1316.05109

[29] Mimura, Masato Sphere equivalence, Banach expanders, and extrapolation, Int. Math. Res. Not. (2015) no. 12, pp. 4372-4391 | Article | MR: 3356758 | Zbl: 1334.46020

[30] Nowak, Piotr W.; Sawicki, Damian Warped cones and spectral gaps, Proc. Am. Math. Soc., Volume 145 (2017) no. 2, pp. 817-823 | Article | MR: 3577880 | Zbl: 1368.46025

[31] Oppenheim, Izhar Averaged projections, angles between groups and strengthening of Banach property (T), Math. Ann., Volume 367 (2017) no. 1-2, pp. 623-666 | Article | MR: 3606450 | Zbl: 06690571

[32] Oppenheim, Izhar Vanishing of cohomology with coefficients in representations on Banach spaces of groups acting on buildings, Comment. Math. Helv., Volume 92 (2017) no. 2, pp. 389-428 | Article | MR: 3656474 | Zbl: 06737226

[33] Reingold, Omer; Vadhan, Salil; Wigderson, Avi Entropy waves, the zig-zag graph product, and new constant-degree expanders, Ann. Math., Volume 155 (2002) no. 1, pp. 157-187 | Article | MR: 1888797 | Zbl: 1008.05101

[34] Roe, John Warped cones and property A, Geom. Topol., Volume 9 (2005), pp. 163-178 | Article | MR: 2115671 | Zbl: 1082.53036

[35] de la Salle, Mikael Towards strong Banach property (T) for SL(3,), Isr. J. Math., Volume 211 (2016) no. 1, pp. 105-145 | Article | MR: 3474958 | Zbl: 1362.46028

[36] Sawicki, Damian Warped cones violating the coarse Baum–Connes conjecture (2017) (Available at www.impan.pl/~dsawicki/)

[37] Sawicki, Damian Warped cones, (non-)rigidity, and piecewise properties, Proc. Lond. Math. Soc., Volume 118 (2019) no. 4, pp. 753-786 (With an appendix by Dawid Kielak and Damian Sawicki) | Article | MR: 3938711 | Zbl: 1417.51011

[38] Sawicki, Damian; Wu, Jianchao Straightening warped cones (2020) (to appear in J. Topol. Anal., https://www.doi.org/10.1142/S179352532050034X)

[39] Vigolo, Federico Discrete fundamental groups of warped cones and expanders, Math. Ann., Volume 373 (2019) no. 1-2, pp. 355-396 | Article | MR: 3968875 | Zbl: 1417.51012

[40] Vigolo, Federico Measure expanding actions, expanders and warped cones, Trans. Am. Math. Soc., Volume 371 (2019) no. 3, pp. 1951-1979 | Article | MR: 3894040 | Zbl: 1402.05205

[41] Willett, Rufus; Yu, Guoliang Higher index theory for certain expanders and Gromov monster groups, I, Adv. Math., Volume 229 (2012) no. 3, pp. 1380-1416 | Article | MR: 2871145 | Zbl: 1243.46060

[42] Yu, Guoliang The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math., Volume 139 (2000) no. 1, pp. 201-240 | Article | MR: 1728880 | Zbl: 0956.19004

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