On the q,p cohomology of Carnot groups
Annales Henri Lebesgue, Volume 1 (2018) , pp. 267-295.

Metadata

Keywords q,p cohomology, Carnot group, global analysis on manifolds

Abstract

We study the simplicial q,p cohomology of Carnot groups G. We show vanishing and non-vanishing results depending of the range of the (p,q) gap with respect to the weight gaps in the Lie algebra cohomology of G.


References

[BFP17] Baldi, Annalisa; Franchi, Bruno; Pansu, Pierre Poincaré and Sobolev inequalities for differential forms in Heisenberg groups (2017) (https://arxiv.org/abs/1711.09786)

[BK15] Bourdon, Marc; Kleiner, Bruce Some applications of p -cohomology to boundaries of Gromov hyperbolic spaces, Groups Geom. Dyn., Volume 9 (2015) no. 2, pp. 435-478 | Article | MR 3356973 | Zbl 1355.20037

[BMOR84] Bemelmans, Josef; Min-Oo, Maung; Ruh, Ernst A. Smoothing Riemannian metrics, Math. Z., Volume 188 (1984) no. 1, pp. 69-74 | Article | MR 767363 | Zbl 0536.53044

[BP03] Bourdon, Marc; Pajot, Hervé Cohomologie l p et espaces de Besov, J. Reine Angew. Math., Volume 558 (2003), pp. 85-108 | Article | MR 1979183 | Zbl 1044.20026

[CGGP92] Christ, Michael; Geller, Daryl; Głowacki, Paweł; Polin, Larry Pseudodifferential operators on groups with dilations, Duke Math. J., Volume 68 (1992) no. 1, pp. 31-65 | Article | MR 1185817 | Zbl 0764.35120

[CT11] Cornulier, Yves; Tessera, Romain Contracting automorphisms and L p -cohomology in degree one, Ark. Mat., Volume 49 (2011) no. 2, pp. 295-324 | Article | MR 2826945 | Zbl 1257.22008

[DM16] Drutu, Cornelia; Mackay, John M. Random groups, random graphs and eigenvalues of p-Laplacians (2016) (https://arxiv.org/abs/1607.04130) | Zbl 06988561

[Ele98] Elek, Gábor Coarse cohomology and l p -cohomology, K-Theory, Volume 13 (1998) no. 1, pp. 1-22 | Article | MR 1610246 | Zbl 0899.46059

[Fol75] Folland, Gerald B. Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., Volume 13 (1975) no. 2, pp. 161-207 | Article | MR 58 | Zbl 0312.35026

[FS74] Folland, Gerald B.; Stein, Elias M. Estimates for the ¯ b complex and analysis on the Heisenberg group, Commun. Pure Appl. Math., Volume 27 (1974), pp. 429-522 | Article | MR 0367477 | Zbl 0293.35012

[Gen14] Genton, Luc Scaled Alexander-Spanier Cohomology and L q,p Cohomology for Metric Spaces (2014) (Thèse no 6330) (Ph. D. Thesis)

[Gro93] 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, 1993, pp. 1-295 | MR 1253544 | Zbl 0841.20039

[Pan08] Pansu, Pierre Cohomologie L p et pincement, Comment. Math. Helv., Volume 83 (2008) no. 2, pp. 327-357 | Article | MR 2390047 | Zbl 1142.53023

[Pan17] Pansu, Pierre Cup-products in L q,p -cohomology: discretization and quasi-isometry invariance (2017) (https://arxiv.org/abs/1702.04984)

[Rum99] Rumin, Michel Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups, C. R. Math. Acad. Sci. Paris, Volume 329 (1999) no. 11, pp. 985-990 | Article

[Rum01] Rumin, Michel Around heat decay on forms and relations of nilpotent Lie groups, Séminaire de Théorie Spectrale et Géométrie, Année 2000–2001 (Séminaire de Théorie Spectrale et Géométrie), Volume 19, Université de Grenoble I, Institut Fourier, 2001, pp. 123-164 | Numdam | MR MR1909080 | Zbl 1035.58021

[Rum10] Rumin, Michel Spectral density and Sobolev inequalities for pure and mixed states, Geom. Funct. Anal., Volume 20 (2010) no. 3, pp. 817-844 | Article | MR 2720233 | Zbl 1218.58021

[VSCC92] Varopoulos, Nicholas Th.; Saloff-Coste, Laurent; Coulhon, Thierry Analysis and geometry on groups, Cambridge Tracts in Mathematics, 100, Cambridge University Press, 1992, xii+156 pages | MR 1218884 | Zbl 1179.22009