[Sur l’annulation de la cohomologie réduite en degré 1 pour les représentations banachiques]
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
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
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.
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Keywords: Property
Mots-clés : Propriété
Cornulier, Yves 1 ; Tessera, Romain 2

@article{AIF_2020__70_5_1951_0, author = {Cornulier, Yves and Tessera, Romain}, title = {On the vanishing of reduced 1-cohomology for {Banach} representations}, journal = {Annales de l'Institut Fourier}, pages = {1951--2003}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {70}, number = {5}, year = {2020}, doi = {10.5802/aif.3363}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3363/} }
TY - JOUR AU - Cornulier, Yves AU - Tessera, Romain TI - On the vanishing of reduced 1-cohomology for Banach representations JO - Annales de l'Institut Fourier PY - 2020 SP - 1951 EP - 2003 VL - 70 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3363/ DO - 10.5802/aif.3363 LA - en ID - AIF_2020__70_5_1951_0 ER -
%0 Journal Article %A Cornulier, Yves %A Tessera, Romain %T On the vanishing of reduced 1-cohomology for Banach representations %J Annales de l'Institut Fourier %D 2020 %P 1951-2003 %V 70 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3363/ %R 10.5802/aif.3363 %G en %F AIF_2020__70_5_1951_0
Cornulier, Yves; Tessera, Romain. On the vanishing of reduced 1-cohomology for Banach representations. Annales de l'Institut Fourier, Tome 70 (2020) no. 5, pp. 1951-2003. doi : 10.5802/aif.3363. https://aif.centre-mersenne.org/articles/10.5802/aif.3363/
[1] General ergodic theorems, Ann. Math., Volume 41 (1940), pp. 293-309 | DOI | MR
[2] Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces, Groups Geom. Dyn., Volume 7 (2013) no. 3, pp. 497-522 | DOI | MR
[3] Property (T) and rigidity for actions on Banach spaces, Acta Math., Volume 198 (2007) no. 1, pp. 57-105 | DOI | MR
[4] On the cohomology of weakly almost periodic group representations, J. Topol. Anal., Volume 6 (2014) no. 2, pp. 153-165 | DOI | MR | Zbl
[5] Kazhdan’s property (T), New Mathematical Monographs, 11, Cambridge University Press, 2008, xiv+472 pages | DOI | MR | Zbl
[6] Analysis on semigroups: Function spaces, compactifications, representations, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, 1989, xiv+334 pages | MR | Zbl
[7] The ergodic theorem for additive cocycles of
[8] The metrical interpretation of superreflexivity in Banach spaces, Isr. J. Math., Volume 56 (1986) no. 2, pp. 222-230 | DOI | MR | Zbl
[9] Shalom’s property
[10] Dimension of asymptotic cones of Lie groups, J. Topol., Volume 1 (2008) no. 2, pp. 342-361 | DOI | MR
[11] Geometric presentations of Lie groups and their Dehn functions, Publ. Math., Inst. Hautes Étud. Sci., Volume 125 (2017), pp. 79-219 | DOI | MR
[12] Isometric group actions on Hilbert spaces: growth of cocycles, Geom. Funct. Anal., Volume 17 (2007) no. 3, pp. 770-792 | DOI | MR
[13]
[14] Operator theoretic aspects of ergodic theory, Graduate Texts in Mathematics, 272, Springer, 2015, xviii+628 pages | DOI | MR
[15] Almost invariance of distributions for random walks on groups, Probab. Theory Relat. Fields, Volume 174 (2019) no. 1-2, pp. 445-476 | DOI | MR | Zbl
[16] Finite-dimensional representations constructed from random walks, Comment. Math. Helv., Volume 93 (2018) no. 3, pp. 555-586 | DOI | MR | Zbl
[17] Isoperimetric inequalities, shapes of Følner sets and groups with Shalom’s property
[18] Random walk in random groups, Geom. Funct. Anal., Volume 13 (2003) no. 1, pp. 73-146 | DOI | MR | Zbl
[19] Sur la cohomologie des groupes topologiques. II, Bull. Sci. Math., Volume 96 (1972), pp. 305-332 | MR | Zbl
[20] Cohomologie des groupes topologiques et des algèbres de Lie, Textes Mathématiques, 2, CEDIC/Fernand Nathan, 1980, xvi+394 pages | MR | Zbl
[21] Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. Fr., Volume 101 (1973), pp. 333-379 | DOI | MR | Zbl
[22] Global existence theorems for harmonic maps to non-locally compact spaces, Commun. Anal. Geom., Volume 5 (1997) no. 2, pp. 333-387 | DOI | MR | Zbl
[23] Cohomological dimension of soluble groups, J. Pure Appl. Algebra, Volume 43 (1986) no. 3, pp. 281-287 | DOI | MR | Zbl
[24] Virtually torsion-free covers of minimax groups, Ann. Sci. Éc. Norm. Supér., Volume 53 (2020) no. 1, pp. 125-171 | DOI | MR | Zbl
[25] On the structure of groups with polynomial growth, Math. Z., Volume 195 (1987) no. 1, pp. 109-117 | DOI | MR
[26] Reduced 1-cohomology of connected locally compact groups and applications, J. Lie Theory, Volume 16 (2006) no. 2, pp. 311-328 | MR
[27] 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
[28]
[29] 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
[30] Rigidity of commensurators and irreducible lattices, Invent. Math., Volume 141 (2000) no. 1, pp. 1-54 | DOI | MR
[31] Harmonic analysis, cohomology, and the large-scale geometry of amenable groups, Acta Math., Volume 192 (2004) no. 2, pp. 119-185 | DOI | MR
[32] Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity, Geom. Funct. Anal., Volume 23 (2013) no. 5, pp. 1631-1683 | DOI | MR
[33] Representations of a compact group on a Banach space, J. Math. Soc. Japan, Volume 7 (1955), pp. 224-248 | DOI | MR
[34] Contractive automorphisms on locally compact groups, Math. Z., Volume 191 (1986) no. 1, pp. 73-90 | DOI | MR | Zbl
[35] Weak Cauchy sequences in
[36] 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] Vanishing of the first reduced cohomology with values in an
[38] Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces, Comment. Math. Helv., Volume 86 (2011) no. 3, pp. 499-535 | DOI | MR | Zbl
[39] 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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