Quasi-isometric invariance of continuous group $L^p$-cohomology, and first applications to vanishings

Bourdon, Marc; Rémy, Bertrand

Annales Henri Lebesgue, Volume 3 (2020), pp. 1291-1326.

We show that the continuous $L^p$-cohomology of locally compact second countable groups is a quasi-isometric invariant. As an application, we prove partial results supporting a positive answer to a question asked by M. Gromov, suggesting a classical behaviour of continuous $L^p$-cohomology of simple real Lie groups. In addition to quasi-isometric invariance, the ingredients are a spectral sequence argument and Pansu’s vanishing results for real hyperbolic spaces. In the best adapted cases of simple Lie groups, we obtain nearly half of the relevant vanishings.

Nous prouvons que la cohomologie $L^p$ continue des groupes localement compacts à base dénombrable d’ouverts est un invariant de quasi-isométrie. En guise d’applications, nous obtenons des résultats partiels dans le sens d’une réponse positive à une question posée par M. Gromov, suggérant un comportement classique de la cohomologie $L^p$ continue des groupes de Lie réels simples. Outre l’invariance par quasi-isométrie, les ingrédients utilisés sont un argument de suite spectrale et des résultats d’annulation pour les espaces hyperboliques dus à P. Pansu. Dans les cas de groupes de Lie les mieux adaptés, nous obtenons la moitié des annulations attendues.

Classification: 20J05, 20J06, 22E15, 22E41, 53C35, 55B35, 57T10, 57T15
Keywords: $L^p$-cohomology, topological group, Lie group, symmetric space, quasi-isometric invariance, spectral sequence, cohomology vanishing, root system

[BFGM07] Bader, Uri; Furman, Alex; Gelander, Tsachik; Monod, Nicolas Property (T) and rigidity for actions on Banach spaces, Acta Math., Volume 198 (2007) no. 1, pp. 57-105 | DOI | MR | Zbl

[BFS14] Bader, Uri; Furman, Alex; Sauer, Roman Weak notions of normality and vanishing up to rank in $L^2$-cohomology, Int. Math. Res. Not. IMRN, Volume 2014 (2014) no. 12, pp. 3177-3189 | DOI | MR | Zbl

[Bla79] Blanc, Philippe Sur la cohomologie continue des groupes localement compacts, Ann. Sci. Éc. Norm. Supér., Volume 12 (1979) no. 2, pp. 137-168 | DOI | Numdam | MR | Zbl

[Bor69] Borel, Armand Introduction aux groupes arithmétiques, Actualités Scientifiques et Industrielles, 1341, Hermann, 1969 | Zbl

[Bor85] Borel, Armand The $L^2$-cohomology of negatively curved Riemennian symmetric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math., Volume 10 (1985), pp. 95-105 | DOI | Zbl

[Bor91] Borel, Armand Linear algebraic groups, Graduate Texts in Mathematics, 126, Springer, 1991 | MR | Zbl

[Bou63] Bourbaki, Nicolas Intégration. Chapitres 7 à 8, Actualités Scientifiques et Industrielles, 1306, Hermann, 1963 | Zbl

[Bou65] Bourbaki, Nicolas Éléments de mathématique. Fasc. XIII. Livre VI :Intégration. Chapitres 1, 2, 3 et 4 : Inégalités de convexité, Espaces de Riesz, Mesures sur les espaces localement compacts, Prolongement d’une mesure, Espaces $L^p$, Actualités Scientifiques et Industrielles, 1175, Hermann, 1965 | Zbl

[Bou68] Bourbaki, Nicolas Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitres IV, V et VI: Groupes de Coxeter et systèmes de Tits. Groupes engendrés par des réflexions. Systèmes de racines, Actualités Scientifiques et Industrielles, 1337, Hermann, 1968 | Zbl

[Bou07] Bourbaki, Nicolas Topologie générale. Chapitres 5 à 10, Springer, 2007 (Reprint of the 1974 original) | Zbl

[Bou16] Bourdon, Marc Cohomologie ${\ell }_p$ en degrés supérieurs et dimension conforme, Ann. Inst. Fourier, Volume 66 (2016) no. 3, pp. 1013-1043 | DOI | MR | Zbl

[BP03] Bourdon, Marc; Pajot, Hervé Cohomologie ${\ell }_p$ et espaces de Besov, J. Reine Angew. Math., Volume 558 (2003), pp. 85-108 | MR | Zbl

[BS87] Burns, Keith; Spatzier, Ralf J. Manifolds of nonpositive curvature and their buildings, Publ. Math. Inst. Hautes Études Sci., Volume 65 (1987), pp. 35-59 | DOI | Numdam | Zbl

[BT65] Borel, Armand; Tits, Jacques Groupes réductifs, Inst. Hautes Études Sci. Publ. Math., Volume 27 (1965), pp. 55-150 | DOI | Numdam

[BW00] Borel, Armand; Wallach, Nolan Continuous cohomology, discrete subgroups, and representations of reductive groups, Mathematical Surveys and Monographs, 67, American Mathematical Society, 2000 | MR

[CdlH16] Cornulier, Yves; de la Harpe, Pierre Metric geometry of locally compact groups, EMS Tracts in Mathematics, 25, European Mathematical Society, 2016 | MR | Zbl

[CT11] Cornulier, Yves; Tessera, Romain Contracting automorphisms and $L^p$-cohomology in degree one, Ark. Mat., Volume 49 (2011) no. 2, pp. 295-324 | DOI | MR | Zbl

[Ele98] Elek, Gábor Coarse cohomology and ${\ell }^p$-cohomology, $K$-Theory, Volume 13 (1998) no. 1, pp. 1-22 | DOI | MR | Zbl

[Fed69] Federer, Herbert Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, 153, Springer, 1969 | MR | Zbl

[Gen14] Genton, Luc Scaled Alexander–Spanier Cohomology and $L^{q,p}$ Cohomology for Metric Spaces, Ph. D. Thesis, EPFL, Lausanne, Switzerland (2014) (These no 6330)

[Gro91] Gromov, Mikhael Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) (London Mathematical Society Lecture Note Series), Volume 182, Cambridge University Press, 1991 | Zbl

[Gui80] Guichardet, Alain Cohomologie des groupes topologiques et des algèbres de Lie, Textes Mathématiques, 2, CEDIC/Fernand Nathan, 1980 | Zbl

[Hei01] Heinonen, Juha Lectures on analysis on metric spaces, Universitext, Springer, 2001 | DOI | Zbl

[Hel01] Helgason, Sigurdur Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, 34, American Mathematical Society, 2001 (Corrected reprint of the 1978 original) | MR | Zbl

[Hum72] Humphreys, James E. Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, 9, Springer, 1972 | MR | Zbl

[Kna02] Knapp, Anthony W. Lie groups beyond an introduction, Progress in Mathematics, 140, Birkhäuser, 2002 | MR | Zbl

[Mau09] Maubon, Julien Symmetric spaces of the non-compact type: differential geometry, Géométries à courbure négative ou nulle, groupes discrets et rigidités (Séminaires et Congrès), Volume 18, Société Mathématique de France, 2009, pp. 1-38 | MR | Zbl

[Mic56] Michael, Ernest Continuous selections I, Ann. Math., Volume 63 (1956), pp. 361-382 | DOI | MR | Zbl

[Mon01] Monod, Nicolas Continuous bounded cohomology of locally compact groups, Lecture Notes in Mathematics, 1758, Springer, 2001 | MR | Zbl

[Mon10] Monod, Nicolas On the bounded cohomology of semi-simple groups, $S$-arithmetic groups and products, J. Reine Angew. Math., Volume 640 (2010) no. 1, pp. 167-202 | MR | Zbl

[Pan89] Pansu, Pierre Cohomologie $L^p$ des variétés à courbure négative, cas du degré $1$, Rend. Sem. Mat. Univ. Politec. Torino, Volume Special Issue (1989), pp. 95-120 Conference on Partial Differential Equations and Geometry (Torino, 1988) | Zbl

[Pan95] Pansu, Pierre Cohomologie $L^p$ : invariance sous quasi isométrie, 1995 (preprint)

[Pan99] Pansu, Pierre Cohomologie $L^p$, espaces homogènes et pincement, 1999 (preprint)

[Pan07] Pansu, Pierre Cohomologie $L^p$ en degré 1 des espaces homogènes, Potential Anal., Volume 27 (2007) no. 2, pp. 151-165 | DOI | Zbl

[Pan08] Pansu, Pierre Cohomologie $L^p$ et pincement, Comment. Math. Helv., Volume 83 (2008) no. 1, pp. 327-357 | DOI | MR | Zbl

[Rez08] Reznikov, Alexander Analytic topology of groups, actions, strings and varieties, Geometry and dynamics of groups and spaces (Progress in Mathematics), Volume 265, Birkhäuser, 2008, pp. 3-93 | DOI | MR | Zbl

[SS18] Sauer, Roman; Schrödl, Michael Vanishing of ${\ell }^2$-Betti numbers of locally compact groups as an invariant of coarse equivalence, Fund. Math., Volume 243 (2018) no. 3, pp. 301-311 | DOI | MR | Zbl

[Tes09] Tessera, Romain Vanishing of the first reduced cohomology with values in a $L^p$-representation, Ann. Inst. Fourier, Volume 59 (2009) no. 2, pp. 851-876 | DOI | MR | Zbl

[Zim84] Zimmer, Robert Ergodic Theory and Semisimple Groups, Monographs in Mathematics, 81, Birkhäuser, 1984 | MR | Zbl