On the vanishing of reduced 1-cohomology for Banach representations  [ Sur l’annulation de la cohomologie réduite en degré 1 pour les représentations banachiques ]
Annales de l'Institut Fourier, à paraître, 53 p.

D’après un théorème de Delorme, pour un groupe de Lie résoluble connexe, toute représentation unitaire dont la 1-cohomologie réduite est non nulle possède une sous-représentation non nulle de dimension finie. Plus récemment, Shalom a démontré que cette propriété passe aux réseaux cocompacts, et est un invariant d’équivalence grossière parmi les groupes discrets moyennables. On donne une nouvelle preuve géométrique du théorème de Delorme, s’étendant à une plus grande classe de groupes, dont les groupes algébriques résolubles p-adiques et les groupes résolubles de type fini de rang de Prüfer fini.

De plus, cela s’applique à des représentations isométriques dans toute une classe d’espaces de Banach, parmi lesquels ceux qui sont réflexifs. On déduit, par exemple, un théorème ergodique pour les cocycles intégrables, ainsi qu’une nouvelle preuve du résultat de Bourgain disant qu’un arbre 3-régulier ne se plonge quasi-isométriquement dans aucun espace de Banach super-réflexif.

A theorem of Delorme states that every unitary representation of a connected solvable Lie group with nontrivial reduced first cohomology has a nonzero finite-dimensional subrepresentation. More recently Shalom showed that such a property is inherited by cocompact lattices and stable under coarse equivalence among amenable countable discrete groups. We give a new geometric proof of Delorme’s theorem, which extends to a larger class of groups, including solvable p-adic algebraic groups and finitely generated solvable groups with finite Prüfer rank.

Moreover all our results apply to isometric representations in a large class of Banach spaces, including reflexive Banach spaces. As applications, we obtain an ergodic theorem in for integrable cocycles, as well as a new proof of Bourgain’s Theorem that the 3-regular tree does not embed quasi-isometrically into any superreflexive Banach space.

Reçu le : 2018-01-15
Révisé le : 2019-04-19
Accepté le : 2019-05-21
Première publication : 2020-10-01
Classification : 43A65,  20F16,  22D10,  22D12,  37A30,  46B99
Mots clés: Propriété fd , Représentations banachiques, Représentations unitaires, Groupes moyennables, Groupes de Lie résolubles, Représentations WAP, Groupes de rang de Prüfer fini
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     author = {Cornulier, Yves and Tessera, Romain},
     title = {On the vanishing of reduced 1-cohomology for Banach representations},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Cornulier, Yves; Tessera, Romain. On the vanishing of reduced 1-cohomology for Banach representations. Annales de l'Institut Fourier, à paraître, 53 p.

[1] Alaoglu, Leonidas; Birkhoff, George D. General ergodic theorems, Ann. Math., Volume 41 (1940), pp. 293-309 | Article | MR 0002026

[2] Austin, Tim; Naor, Assaf; Tessera, Romain Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces, Groups Geom. Dyn., Volume 7 (2013) no. 3, pp. 497-522 | Article | MR 3095705

[3] 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

[4] Bader, Uri; Rosendal, Christian; Sauer, Roman On the cohomology of weakly almost periodic group representations, J. Topol. Anal., Volume 6 (2014) no. 2, pp. 153-165 | Article | MR 3191647

[5] 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

[6] Berglund, John F.; Junghenn, Hugo D.; Milnes, Paul Analysis on semigroups: Function spaces, compactifications, representations, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, 1989, xiv+334 pages | MR 999922 | Zbl 0727.22001

[7] Boivin, Daniel; Derriennic, Yves The ergodic theorem for additive cocycles of Z d or R d , Ergodic Theory Dyn. Syst., Volume 11 (1991) no. 1, pp. 19-39 | Article | MR 1101082

[8] Bourgain, Jean The metrical interpretation of superreflexivity in Banach spaces, Isr. J. Math., Volume 56 (1986) no. 2, pp. 222-230 | Article | MR 880292

[9] Brieussel, Jérémie; Zheng, Tianyi Shalom’s property H FD and extensions by of locally finite groups, Isr. J. Math., Volume 230 (2019) no. 1, pp. 45-70 | Article | MR 3941140

[10] de Cornulier, Yves Dimension of asymptotic cones of Lie groups, J. Topol., Volume 1 (2008) no. 2, pp. 342-361 | Article | MR 2399134

[11] Cornulier, Yves; Tessera, Romain Geometric presentations of Lie groups and their Dehn functions, Publ. Math., Inst. Hautes Étud. Sci., Volume 125 (2017), pp. 79-219 | Article | MR 3668649

[12] de Cornulier, Yves; Tessera, Romain; Valette, Alain Isometric group actions on Hilbert spaces: growth of cocycles, Geom. Funct. Anal., Volume 17 (2007) no. 3, pp. 770-792 | Article | MR 2346274

[13] 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 | Article | MR 0578893

[14] Eisner, Tanja; Farkas, Bálint; Haase, Markus; Nagel, Rainer Operator theoretic aspects of ergodic theory, Graduate Texts in Mathematics, Volume 272, Springer, 2015, xviii+628 pages | Article | MR 3410920

[15] Erschler, Anna Almost invariance of distributions for random walks on groups, Probab. Theory Relat. Fields, Volume 174 (2019) no. 1-2, pp. 445-476 | Article | MR 3947328

[16] Erschler, Anna; Ozawa, Narutaka Finite-dimensional representations constructed from random walks, Comment. Math. Helv., Volume 93 (2018) no. 3, pp. 555-586 | Article | MR 3854902

[17] Erschler, Anna; Zheng, Tianyi Isoperimetric inequalities, shapes of Følner sets and groups with Shalom’s property H FD (2017) (https://arxiv.org/abs/1708.04730, to appear in Ann. Inst. Fourier)

[18] Gromov, Mikhail Random walk in random groups, Geom. Funct. Anal., Volume 13 (2003) no. 1, pp. 73-146 | Article | MR 1978492

[19] Guichardet, Alain Sur la cohomologie des groupes topologiques. II, Bull. Sci. Math., Volume 96 (1972), pp. 305-332 | MR 0340464

[20] Guichardet, Alain Cohomologie des groupes topologiques et des algèbres de Lie, Textes Mathématiques, Volume 2, CEDIC/Fernand Nathan, 1980, xvi+394 pages | MR 644979

[21] Guivarc’h, Yves Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. Fr., Volume 101 (1973), pp. 333-379 | Article | MR 0369608

[22] Korevaar, Nicholas J.; Schoen, Richard M. Global existence theorems for harmonic maps to non-locally compact spaces, Commun. Anal. Geom., Volume 5 (1997) no. 2, pp. 333-387 | Article | MR 1483983

[23] Kropholler, Peter H. Cohomological dimension of soluble groups, J. Pure Appl. Algebra, Volume 43 (1986) no. 3, pp. 281-287 | Article | MR 868988

[24] Kropholler, Peter H.; Lorensen, Karl Virtually torsion-free covers of minimax groups, Ann. Sci. Éc. Norm. Supér., Volume 53 (2020) no. 1, pp. 125-171 | Zbl 07201738

[25] Losert, Viktor On the structure of groups with polynomial growth, Math. Z., Volume 195 (1987) no. 1, pp. 109-117 | Article | MR 888132

[26] Martin, Florian Reduced 1-cohomology of connected locally compact groups and applications, J. Lie Theory, Volume 16 (2006) no. 2, pp. 311-328 | MR 2197595

[27] Mok, Ngaiming Harmonic forms with values in locally constant Hilbert bundles, Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993) (1995), pp. 433-453 | MR 1364901

[28] Naor, Assaf; Peres, Yuval L p compression, traveling salesmen, and stable walks, Duke Math. J., Volume 157 (2011) no. 1, pp. 53-108 | Article | MR 2783928

[29] 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 | Article | MR 3831031

[30] Shalom, Yehuda Rigidity of commensurators and irreducible lattices, Invent. Math., Volume 141 (2000) no. 1, pp. 1-54 | Article | MR 1767270

[31] Shalom, Yehuda Harmonic analysis, cohomology, and the large-scale geometry of amenable groups, Acta Math., Volume 192 (2004) no. 2, pp. 119-185 | Article | MR 2096453

[32] Shalom, Yehuda; Willis, George A. Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity, Geom. Funct. Anal., Volume 23 (2013) no. 5, pp. 1631-1683 | Article | MR 3102914

[33] Shiga, Kôji Representations of a compact group on a Banach space, J. Math. Soc. Japan, Volume 7 (1955), pp. 224-248 | Article | MR 0082624

[34] Siebert, Eberhard Contractive automorphisms on locally compact groups, Math. Z., Volume 191 (1986) no. 1, pp. 73-90 | Article | MR 812604

[35] Talagrand, Michel Weak Cauchy sequences in L 1 (E), Am. J. Math., Volume 106 (1984) no. 3, pp. 703-724 | Article | MR 745148

[36] Tessera, Romain Large scale Sobolev inequalities on metric measure spaces and applications, Rev. Mat. Iberoam., Volume 24 (2008) no. 3, pp. 825-864 | Article | MR 2490163

[37] Tessera, Romain Vanishing of the first reduced cohomology with values in an L p -representation, Ann. Inst. Fourier, Volume 59 (2009) no. 2, pp. 851-876 | Article | MR 2521437 | Zbl 1225.22019

[38] Tessera, Romain Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces, Comment. Math. Helv., Volume 86 (2011) no. 3, pp. 499-535 | Article | MR 2803851

[39] Tessera, Romain Isoperimetric profile and random walks on locally compact solvable groups, Rev. Mat. Iberoam., Volume 29 (2013) no. 2, pp. 715-737 | Article | MR 3047434