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

Bourdon, Marc; Rémy, Bertrand

doi:10.5802/ahl.61

Bourdon, Marc; Rémy, Bertrand

Quasi-isometric invariance of continuous group

L^{p}

-cohomology, and first applications to vanishings

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

Metadata

DOI10.5802/ahl.61

Keywords $L^p$-cohomology, topological group, Lie group, symmetric space, quasi-isometric invariance, spectral sequence, cohomology vanishing, root system

Abstract

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.

References

[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 $ℓ_{p}$ 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 $ℓ_{p}$ 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 $ℓ^{p}$ -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 $ℓ^{2}$ -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

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