# ANNALES DE L'INSTITUT FOURIER

Analytic properties of approximate lattices  [ Propriétés analytiques de réseaux approximatifs ]
Annales de l'Institut Fourier, à paraître, 48 p.

Nous introduisons une notion d’induction de cocycle pour les réseaux approximatifs uniformes forts dans les groupes localement compacts à base dénombrable, et nous l’utilisons pour mettre en relation les réseaux approximatifs de type Kazhdan et Haagerup (relatifs) avec les propriétés correspondantes des groupes ambiants localement compacts. Notre approche s’applique à de larges classes de réseaux approximatifs uniformes (bien que pas toutes) et est suffisamment souple pour couvrir les versions ${L}^{p}$ de propriété (FH) et a-(FH)-moyennabilité, ainsi que leurs versions quasi à la Burger–Monod et Ozawa.

We introduce a notion of cocycle-induction for strong uniform approximate lattices in locally compact second countable groups and use it to relate (relative) Kazhdan- and Haagerup-type of approximate lattices to the corresponding properties of the ambient locally compact groups. Our approach applies to large classes of uniform approximate lattices (though not all of them) and is flexible enough to cover the ${L}^{p}$-versions of Property (FH) and a-(FH)-menability as well as quasified versions thereof a la Burger–Monod and Ozawa.

Reçu le : 2017-10-13
Révisé le : 2018-09-27
Accepté le : 2019-05-21
Première publication : 2020-10-01
Classification : 20N99,  22D10,  22E40
Mots clés: Réseaux approximatifs, propriété (T), propriét é (FH), propriété Haagerup
@unpublished{AIF_0__0_0_A19_0,
author = {Bj\"orklund, Michael and Hartnick, Tobias},
title = {Analytic properties of approximate lattices},
note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Björklund, Michael; Hartnick, Tobias. Analytic properties of approximate lattices. Annales de l'Institut Fourier, à paraître, 48 p.

[1] Bader, Uri; Furman, Alex; Gelander, Tsachik; Monod, Nicolas Property (T) and rigidity for actions on Banach spaces, Acta Math., Volume 198 (2007) no. 1, pp. 57-105 | Article | MR 2316269

[2] Bekka, Bachir; de la Harpe, Pierre; Valette, Alain Kazhdan’s property (T), New Mathematical Monographs, Volume 11, Cambridge University Press, 2008, xiv+472 pages | Article | MR 2415834

[3] Benedetti, Riccardo; Petronio, Carlo Lectures on hyperbolic geometry, Universitext, Springer, 1992, xiv+330 pages | Article | MR 1219310

[4] Björklund, Michael; Hartnick, Tobias Approximate lattices, Duke Math. J., Volume 167 (2018) no. 15, pp. 2903-2964 | Article | MR 3865655

[5] Björklund, Michael; Hartnick, Tobias; Pogorzelski, Felix Aperiodic order and spherical diffraction, I: auto-correlation of regular model sets, Proc. Lond. Math. Soc., Volume 116 (2018) no. 4, pp. 957-996 | Article | MR 3789837

[6] Burger, Marc; Monod, Nicolas Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal., Volume 12 (2002) no. 2, pp. 219-280 | Article | MR 1911660

[7] Cherix, Pierre-Alain; Cowling, Michael; Jolissaint, Paul; Julg, Pierre; Valette, Alain Groups with the Haagerup property. Gromov’s a-T-menability, Progress in Mathematics, Volume 197, Birkhäuser, 2001, viii+126 pages | Article | MR 1852148

[8] Chifan, Ionut; Ioana, Adrian On relative property (T) and Haagerup’s property, Trans. Am. Math. Soc., Volume 363 (2011) no. 12, pp. 6407-6420 | Article | MR 2833560

[9] de Cornulier, Yves On Haagerup and Kazhdan properties. (2006) (Ph. D. Thesis)

[10] de Cornulier, Yves Relative Kazhdan property, Ann. Sci. Éc. Norm. Supér., Volume 39 (2006) no. 2, pp. 301-333 | Article | MR 2245534

[11] De Cornulier, Yves; Tessera, Romain; Valette, Alain Isometric group actions on Banach spaces and representations vanishing at infinity, Transform. Groups, Volume 13 (2008) no. 1, pp. 125-147 | Article | MR 2421319

[12] Gromov, Misha Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) (London Mathematical Society Lecture Note Series) Volume 182, Cambridge University Press, 1993, pp. 1-295 | MR 1253544

[13] Haagerup, Uffe An example of a nonnuclear ${C}^{*}$-algebra, which has the metric approximation property, Invent. Math., Volume 50 (1978/79) no. 3, pp. 279-293 | Article | MR 520930

[14] Každan, David A. On the connection of the dual space of a group with the structure of its closed subgroups, Funkts. Anal. Prilozh., Volume 1 (1967), pp. 71-74 | MR 0209390

[15] Margulis, Grigoriĭ A. Finitely-additive invariant measures on Euclidean spaces, Ergodic Theory Dyn. Syst., Volume 2 (1982) no. 3-4, pp. 383-396 | Article | MR 721730

[16] Meyer, Yves Algebraic numbers and harmonic analysis, North-Holland Mathematical Library, Volume 2, North-Holland; Elsevier, 1972, x+274 pages | MR 0485769

[17] Mimura, Masato Fixed point properties and second bounded cohomology of universal lattices on Banach spaces, J. Reine Angew. Math., Volume 653 (2011), pp. 115-134 | Article | MR 2794627

[18] Monod, Nicolas Continuous bounded cohomology of locally compact groups, Lecture Notes in Mathematics, Volume 1758, Springer, 2001, x+214 pages | Article | MR 1840942

[19] Montgomery, Deane; Zippin, Leo Topological transformation groups, Interscience Publishers, 1955, xi+282 pages | MR 0073104

[20] Moody, Robert V. Meyer sets and their duals, The mathematics of long-range aperiodic order (Waterloo, ON, 1995) (NATO ASI Series. Series C. Mathematical and Physical Sciences) Volume 489, Kluwer Academic Publishers, 1997, pp. 403-441 | MR 1460032

[21] Ozawa, Narutaka Quasi-homomorphism rigidity with non-commutative targets, J. Reine Angew. Math., Volume 655 (2011), pp. 89-104 | Article | MR 2806106

[22] Pettis, Billy J. On continuity and openness of homomorphisms in topological groups, Ann. Math., Volume 52 (1950), pp. 293-308 | Article | MR 0038358

[23] Shalom, Yehuda Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group, Ann. Math., Volume 152 (2000) no. 1, pp. 113-182 | Article | MR 1792293

[24] Tao, Terence Product set estimates for non-commutative groups, Combinatorica, Volume 28 (2008) no. 5, pp. 547-594 | Article | MR 2501249