On a variant of Kazhdan's property (T) for subgroups of semisimple groups
Annales de l'Institut Fourier, Volume 47 (1997) no. 4, pp. 1065-1078.

Let Γ be an irreducible lattice in a product G of simple groups. Assume that G has a factor with property (T). We give a description of the topology in a neighbourhood of the trivial one dimensional representation of Γ in terms of the topology of the dual space G ^ of G.

We use this result to give a new proof for the triviality of the first cohomology group of Γ with coefficients in a finite dimensional unitary representation.

Soit Γ un réseau irréductible dans un produit G de groupes simples. Supposons qu’un des facteurs de G possède la propriété (T). Nous donnons une description de la topologie dans un voisinage de la représentation triviale de dimension un de Γ en termes de celle du dual G ^ de G .

Nous utilisons ce résultat pour donner une nouvelle preuve de l’annulation du premier groupe de cohomologie de Γ à coefficients dans une représentation unitaire de dimension finie.

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     author = {Bekka, Mohammed Bachir and Louvet, Nicolas},
     title = {On a variant of {Kazhdan's} property {(T)} for subgroups of semisimple groups},
     journal = {Annales de l'Institut Fourier},
     pages = {1065--1078},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {4},
     year = {1997},
     doi = {10.5802/aif.1591},
     mrnumber = {99b:22019},
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     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1591/}
}
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Bekka, Mohammed Bachir; Louvet, Nicolas. On a variant of Kazhdan's property (T) for subgroups of semisimple groups. Annales de l'Institut Fourier, Volume 47 (1997) no. 4, pp. 1065-1078. doi : 10.5802/aif.1591. http://archive.numdam.org/articles/10.5802/aif.1591/

[Bor] A. Borel, Some finiteness properties of adèles groups over numbers fields, Publ. Math. IHES, 16 (1963), 1-30. | Numdam | Zbl

[BoW] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups and representations of reductive groups, Annals of Math. Studies, Princeton University Press, 1980. | Zbl

[Del] P. Delorme, 1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles Produits tensoriels continus de représentations, Bull. Soc. Math. France, 105 (1977), 281-336. | Numdam | MR | Zbl

[Dix] J. Dixmier, Les C*-algèbres et leurs représentations, Gauthier-Villars, 1969. | MR | Zbl

[Fe1] J.M.G. Fell, Weak containment and induced representations of groups, Canad. J. Math., 14 (1962), 237-268. | MR | Zbl

[Fe2] J.M.G. Fell, Weak containment and Kronecker products of group representations, Pac. J. Math., 13 (1963), 503-510. | MR | Zbl

[Gu1] A. Guichardet, Symmetric Hilbert spaces and related topics, Lecture Notes in Math., 261, Springer, 1972. | Zbl

[Gu2] A. Guichardet, Cohomologie des groupes localement compacts et produits tensoriels continus de représentations, J. Multivariate Anal., 6 (1976), 138-158. | MR | Zbl

[Gu3] A. Guichardet, Tensor products of C*-algebras, Part II, Lecture Notes Series, 13, Aarhus Universitet, 1969. | Zbl

[Hum] J.E. Humphreys, Arithmetic groups, Lecture Notes in Math., 789, Springer, 1980. | MR | Zbl

[HaV] P. De La Harpe et A. Valette, La propriété (T) de Kazhdan pour les groupes localement compacts, Astérisque 75, Soc. Math. de France, 1989. | Numdam | Zbl

[Lub] A. Lubotzky, Discrete groups, expanding graphs and invariant measures, Progress in Math. 125, Birkhäuser, 1994. | MR | Zbl

[LuZ] A. Lubotzky and R.J. Zimmer, Variants of Kazhdan's property for subgroup of semisimple groups, Israel J. of Math., 66 (1989), 289-298. | MR | Zbl

[Mar] G.A. Margulis, Discrete subgroups of semisimple Lie groups, Springer, 1991. | MR | Zbl

[VeK] A.M. Vershik and S.I. Karpushev, Cohomology of groups in unitary representations, the neighbourhood of the identity and conditionally positive definite functions, Math. USSR Sbornik, 47 (1984), 513-526. | Zbl

[Zim] R.J. Zimmer, Ergodic theory and semisimple groups, Birkhäuser, Boston, 1984. | MR | Zbl

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