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 -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.
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 -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.
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Accepted:
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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
@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, Volume 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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