[Non-annulation de la cohomologie pour les groupes résolubles et semi-simples]
Nous obtenons des résultats de non-annulation de la cohomologie pour des groupes de Lie lorsque est grand et quand le degré est égal au rang du groupe. Ces résultats s’appliquent à la fois aux groupes semi-simples et à certains groupes résolubles. En particulier, ils confirment que la question de Gromov concernant l’annulation en-dessous du rang est formulée de façon optimale. Pour obtenir ces résultats, des annulations complémentaires sont combinées à l’usage de suites spectrales. Afin de déduire le cas semi-simple du cas résoluble, nous utilisons également des comparaisons entre diverses versions de la cohomologie , et nous appliquons l’invariance par quasi-isométrie.
We obtain non-vanishing of group -cohomology of Lie groups for large and when the degree is equal to the rank of the group. This applies both to semisimple and to some suitable solvable groups. In particular, it confirms that Gromov’s question on vanishing below the rank is formulated optimally. To achieve this, some complementary vanishings are combined with the use of spectral sequences. To deduce the semisimple case from the solvable one, we also need comparison results between various theories for -cohomology, allowing the use of quasi-isometry invariance.
Accepté le :
Publié le :
Keywords: $L^p$-cohomology, Lie group, symmetric space, quasi-isometric invariance, spectral sequence, cohomology (non-)vanishing, root system
Mot clés : Cohomologie $L^p$, groupe de Lie, espace symétrique, invariance par quasi-isométrie, suite spectrale, annulation et non-annulation de cohomologie, système de racines
Marc Bourdon 1 ; Bertrand Rémy 2
CC-BY 4.0
@article{JEP_2023__10__771_0,
author = {Marc Bourdon and Bertrand R\'emy},
title = {Non-vanishing for group $L^p$-cohomology of solvable and semisimple {Lie} groups},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {771--814},
publisher = {\'Ecole polytechnique},
volume = {10},
year = {2023},
doi = {10.5802/jep.232},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.232/}
}
TY - JOUR AU - Marc Bourdon AU - Bertrand Rémy TI - Non-vanishing for group $L^p$-cohomology of solvable and semisimple Lie groups JO - Journal de l’École polytechnique — Mathématiques PY - 2023 SP - 771 EP - 814 VL - 10 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.232/ DO - 10.5802/jep.232 LA - en ID - JEP_2023__10__771_0 ER -
%0 Journal Article %A Marc Bourdon %A Bertrand Rémy %T Non-vanishing for group $L^p$-cohomology of solvable and semisimple Lie groups %J Journal de l’École polytechnique — Mathématiques %D 2023 %P 771-814 %V 10 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.232/ %R 10.5802/jep.232 %G en %F JEP_2023__10__771_0
Marc Bourdon; Bertrand Rémy. Non-vanishing for group $L^p$-cohomology of solvable and semisimple Lie groups. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 771-814. doi : 10.5802/jep.232. https://jep.centre-mersenne.org/articles/10.5802/jep.232/
[AW76] - “Homogeneous manifolds with negative curvature. I”, Trans. Amer. Math. Soc. 215 (1976), p. 323-362 | DOI | MR | Zbl
[BFGM07] - “Property (T) and rigidity for actions on Banach spaces”, Acta Math. 198 (2007) no. 1, p. 57-105 | DOI | MR | Zbl
[BFS14] - “Weak notions of normality and vanishing up to rank in -cohomology”, Internat. Math. Res. Notices (2014) no. 12, p. 3177-3189 | DOI | MR | Zbl
[Bor85] - “The -cohomology of negatively curved Riemannian symmetric spaces”, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), p. 95-105 | DOI | MR | Zbl
[Bou68] - Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitres IV à VI, Actualités Scientifiques et Industrielles, vol. 1337, Hermann, Paris, 1968
[BR20] - “Quasi-isometric invariance of continuous group -cohomology, and first applications to vanishings”, Ann. H. Lebesgue 3 (2020), p. 1291-1326 | DOI | MR | Zbl
[BT82] - Differential forms in algebraic topology, Graduate Texts in Math., vol. 82, Springer-Verlag, New York-Berlin, 1982 | DOI
[BW00] - Continuous cohomology, discrete subgroups, and representations of reductive groups, Math. Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000 | DOI
[CdlH16] - Metric geometry of locally compact groups, EMS Tracts in Math., vol. 25, European Mathematical Society, Zürich, 2016 | DOI
[Cor08] - “Dimension of asymptotic cones of Lie groups”, J. Topology 1 (2008) no. 2, p. 342-361 | DOI | MR | Zbl
[CT11] - “Contracting automorphisms and -cohomology in degree one”, Ark. Mat. 49 (2011) no. 2, p. 295-324 | DOI | MR | Zbl
[DS05] - “Introduction to the theory of currents”, 2005, Available at https://webusers.imj-prg.fr/~tien-cuong.dinh/Cours2005/Master/cours.pdf
[Ele98] - “Coarse cohomology and -cohomology”, -Theory 13 (1998) no. 1, p. 1-22 | DOI | MR | Zbl
[Gen14] - Scaled Alexander-Spanier cohomology and cohomology for metric spaces, Ph. D. Thesis, EPFL, Lausanne, 2014, These no. 6330
[GHL04] - Riemannian geometry, Universitext, Springer-Verlag, Berlin, 2004 | DOI
[Gro93] - “Asymptotic invariants of infinite groups”, in Geometric group theory, Vol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser., vol. 182, Cambridge University Press, Cambridge, 1993, p. 1-295 | MR
[GT06] - “Sobolev inequalities for differential forms and -cohomology”, J. Geom. Anal. 16 (2006) no. 4, p. 597-631 | DOI | MR | Zbl
[GT10] - “A short proof of the Hölder-Poincaré duality for -cohomology”, Rend. Sem. Mat. Univ. Padova 124 (2010), p. 179-184 | DOI | Zbl
[Gui80] - Cohomologie des groupes topologiques et des algèbres de Lie, Textes Mathématiques, vol. 2, CEDIC, Paris, 1980
[Hei74] - “On homogeneous manifolds of negative curvature”, Math. Ann. 211 (1974), p. 23-34 | DOI | MR | Zbl
[Hel01] - Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Math., vol. 34, American Mathematical Society, Providence, RI, 2001 | DOI
[HS53] - “Cohomology of group extensions”, Trans. Amer. Math. Soc. 74 (1953), p. 110-134 | DOI | MR | Zbl
[Hum78] - Introduction to Lie algebras and representation theory, Graduate Texts in Math., vol. 9, Springer-Verlag, New York-Berlin, 1978
[IL93] - “Integral estimates for null Lagrangians”, Arch. Rational Mech. Anal. 125 (1993) no. 1, p. 25-79 | DOI | MR | Zbl
[Pan95] - “Cohomologie : invariance sous quasiisométrie”, 1995, Preprint available at https://www.imo.universite-paris-saclay.fr/~pierre.pansu/liste-prepub.html
[Pan99] - “Cohomologie , espaces homogènes et pincement”, 1999, Preprint available at https://www.imo.universite-paris-saclay.fr/~pierre.pansu/liste-prepub.html
[Pan07] - “Cohomologie en degré des espaces homogènes”, Potential Anal. 27 (2007) no. 2, p. 151-165 | DOI | Zbl
[Pan08] - “Cohomologie et pincement”, Comment. Math. Helv. 83 (2008) no. 2, p. 327-357 | DOI | MR | Zbl
[Pan09] - “Pincement du plan hyperbolique complexe”, 2009, Preprint available at https://www.imo.universite-paris-saclay.fr/~pierre.pansu/liste-prepub.html
[Seq20] - Relative and Orlicz cohomology and applications to Heintze groups, Ph. D. Thesis, Universidad de la República Uruguay and Université de Lille, 2020
[SS18] - “Vanishing of -Betti numbers of locally compact groups as an invariant of coarse equivalence”, Fund. Math. 243 (2018) no. 3, p. 301-311 | DOI | MR | Zbl
[Tes09] - “Vanishing of the first reduced cohomology with values in an -representation”, Ann. Inst. Fourier (Grenoble) 59 (2009) no. 2, p. 851-876 | DOI | MR | Zbl
[Var96] - “Analysis on Lie groups”, Rev. Mat. Iberoamericana 12 (1996) no. 3, p. 791-917 | DOI | MR | Zbl
Cité par Sources :