Les moyennes sont des invariants definis sur la cohomologie des groupes de Lie. Nous démontrons que les moyennes dans les groupes abéliens et les groupes d’Heisenberg sont nulles. Ce résultat complète des travaux précédents et montre que la cohomologie est nulle pour les groupes de Lie étudiés.
Averages are invariants defined on the cohomology of Lie groups. We prove that they vanish for abelian and Heisenberg groups. This result completes work by other authors and allows to show that the cohomology vanishes in these cases.
@article{AMBP_2019__26_1_81_0, author = {Pierre Pansu and Francesca Tripaldi}, title = {Averages and the $\ell ^{q,1}$ cohomology of Heisenberg groups}, journal = {Annales Math\'ematiques Blaise Pascal}, pages = {81--100}, publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal}, volume = {26}, number = {1}, year = {2019}, doi = {10.5802/ambp.384}, language = {en}, url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.384/} }
Pierre Pansu; Francesca Tripaldi. Averages and the $\ell ^{q,1}$ cohomology of Heisenberg groups. Annales Mathématiques Blaise Pascal, Tome 26 (2019) no. 1, pp. 81-100. doi : 10.5802/ambp.384. https://ambp.centre-mersenne.org/articles/10.5802/ambp.384/
[1] Poincaré and Sobolev inequalities for differential forms in Heisenberg groups (2017) (https://arxiv.org/abs/1711.09786)
[2] -Poincaré inequalities for differential forms on Euclidean spaces and Heisenberg groups (2019) (https://arxiv.org/abs/1902.04819)
[3] Compensated compactness in the contact complex of Heisenberg groups, Indiana Univ. Math. J., Volume 57 (2008) no. 1, pp. 133-185 | Article | MR 2400254 | Zbl 1143.43007
[4] New estimates for elliptic equations and Hodge type systems, J. Eur. Math. Soc., Volume 009 (2007) no. 2, pp. 277-315 | Article | MR 2293957 | Zbl 1176.35061
[5] Subelliptic Bourgain–Brezis estimates on groups, Math. Res. Lett., Volume 16 (2009) no. 2-3, pp. 487-501 | Article | MR 2511628 | Zbl 1184.35119
[6] Cup-products in -cohomology: discretization and quasi-isometry invariance (2017) (https://arxiv.org/abs/1702.04984)
[7] On the cohomology of Carnot groups (to appear in Ann. Henri Lebesgue) | Zbl 07099970
[8] The connectivity at infinity of a manifold and -Sobolev inequalities, Expo. Math., Volume 32 (2014) no. 4, pp. 365-383 | Article | MR 3279484 | Zbl 1303.53055
[9] Differential forms on contact manifolds, J. Differ. Geom., Volume 39 (1994) no. 2, pp. 281-330 | MR 1267892 | Zbl 0973.53524