no. 5
On the vanishing of reduced 1-cohomology for Banach representations
[Sur l’annulation de la cohomologie réduite en degré 1 pour les représentations banachiques]
Cornulier, Yves1 ; Tessera, Romain2
1 CNRS and Univ Lyon, Univ. Claude Bernard Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, 69622 Villeurbanne (France)
2 CNRS, Institut de Mathématiques de Jussieu – Paris Rive Gauche (IMJ-PRG), Campus des Grands Moulins Université Paris Diderot – Boite Courrier 7012, 8 Place Aurélie Nemours, 75205 Paris Cedex 13 (France)
Annales de l'Institut Fourier, Tome 70 (2020) no. 5, pp. 1951-2003.

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.

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DOI : 10.5802/aif.3363
Classification : 43A65, 20F16, 22D10, 22D12, 37A30, 46B99
Keywords: Property $\mathcal{H}_{\mathrm{fd}}$, Banach representations, Unitary representations, Amenable groups, Solvable Lie groups, WAP representations, Groups of finite Prüfer rank
Mot clés : Propriété $\mathcal{H}_{\mathrm{fd}}$, Représentations banachiques, Représentations unitaires, Groupes moyennables, Groupes de Lie résolubles, Représentations WAP, Groupes de rang de Prüfer fini
Cornulier, Yves 1 ; Tessera, Romain 2

1 CNRS and Univ Lyon, Univ. Claude Bernard Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, 69622 Villeurbanne (France)
2 CNRS, Institut de Mathématiques de Jussieu – Paris Rive Gauche (IMJ-PRG), Campus des Grands Moulins Université Paris Diderot – Boite Courrier 7012, 8 Place Aurélie Nemours, 75205 Paris Cedex 13 (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {On the vanishing of reduced 1-cohomology for {Banach} representations},
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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/

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