Cohomologie p en degrés supérieurs et dimension conforme  [  p -Cohomology in higher degrees and conformal dimension ]
Annales de l'Institut Fourier, Volume 66 (2016) no. 3, p. 1013-1043

We establish a sufficient condition for vanishing of the p -cohomology of hyperbolic uniformly contractible simplicial complexes, in degree at least 2. As an application, a geometric lower bound for the conformal dimension of the boundary at infinity of some hyperbolic groups, is obtained.

On donne une condition suffisante pour l’annulation de la cohomologie p en degré supérieur à 2, des complexes simpliciaux hyperboliques uniformément contractiles. Comme application, on obtient une minoration de la dimension conforme du bord à l’infini de certains groupes hyperboliques.

Received : 2015-01-15
Revised : 2015-07-06
Accepted : 2015-09-10
Published online : 2016-12-14
DOI : https://doi.org/10.5802/aif.3030
Classification:  20F65,  20F67,  30L10
Keywords: p -cohomology, conformal dimension, hyperbolic spaces
@article{AIF_2016__66_3_1013_0,
     author = {Bourdon, Marc},
     title = {Cohomologie $\ell \_p$ en degr\'es sup\'erieurs et dimension conforme},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {3},
     year = {2016},
     pages = {1013-1043},
     doi = {10.5802/aif.3030},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2016__66_3_1013_0}
}
Bourdon, Marc. Cohomologie $\ell _p$ en degrés supérieurs et dimension conforme. Annales de l'Institut Fourier, Volume 66 (2016) no. 3, pp. 1013-1043. doi : 10.5802/aif.3030. http://www.numdam.org/item/AIF_2016__66_3_1013_0/

[1] Assouad, Patrice Plongements lipschitziens dans R n , Bull. Soc. Math. France, Tome 111 (1983) no. 4, pp. 429-448

[2] Bestvina, Mladen; Mess, Geoffrey The boundary of negatively curved groups, J. Amer. Math. Soc., Tome 4 (1991) no. 3, pp. 469-481 | Article

[3] Bonk, M.; Schramm, O. Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal., Tome 10 (2000) no. 2, pp. 266-306 | Article

[4] Bonk, Mario; Heinonen, Juha; Koskela, Pekka Uniformizing Gromov hyperbolic spaces, Astérisque (2001) no. 270, viii+99 pages

[5] Bonk, Mario; Kleiner, Bruce Rigidity for quasi-Möbius group actions, J. Differential Geom., Tome 61 (2002) no. 1, pp. 81-106 http://projecteuclid.org/euclid.jdg/1090351321

[6] Bourdon, Marc Structure conforme au bord et flot géodésique d’un CAT (-1)-espace, Enseign. Math. (2), Tome 41 (1995) no. 1-2, pp. 63-102

[7] Bourdon, Marc Cohomologie l p et produits amalgamés, Geom. Dedicata, Tome 107 (2004), pp. 85-98 | Article

[8] Bourdon, Marc Cohomologie et actions isométriques propres sur les espaces L p , Geometry, Topology, and Dynamics in Negative Curvature, (Bangalore 2010), London Math. Soc. Lecture Note Ser., vol. 425, Cambridge Univ. Press (2016) (à paraître)

[9] Bourdon, Marc; Kleiner, Bruce Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups, Groups Geom. Dyn., Tome 7 (2013) no. 1, pp. 39-107 | Article

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

[11] Bourdon, Marc; Pajot, Hervé Cohomologie l p et espaces de Besov, J. Reine Angew. Math., Tome 558 (2003), pp. 85-108 | Article

[12] Canary, R. D.; Epstein, D. B. A.; Green, P. Notes on notes of Thurston, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 111 (1987), pp. 3-92

[13] Carrasco Piaggio, Matias Conformal dimension and canonical splittings of hyperbolic groups, Geom. Funct. Anal., Tome 24 (2014) no. 3, pp. 922-945 | Article

[14] Coornaert, Michel Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov, Pacific J. Math., Tome 159 (1993) no. 2, pp. 241-270 http://projecteuclid.org/euclid.pjm/1102634263

[15] Dodziuk, Jozef de Rham-Hodge theory for L 2 -cohomology of infinite coverings, Topology, Tome 16 (1977) no. 2, pp. 157-165

[16] Donnelly, Harold; Xavier, Frederico On the differential form spectrum of negatively curved Riemannian manifolds, Amer. J. Math., Tome 106 (1984) no. 1, pp. 169-185 | Article

[17] Elek, Gábor The l p -cohomology and the conformal dimension of hyperbolic cones, Geom. Dedicata, Tome 68 (1997) no. 3, pp. 263-279 | Article

[18] Elek, Gábor Coarse cohomology and l p -cohomology, K-Theory, Tome 13 (1998) no. 1, pp. 1-22 | Article

[19] Gersten, S. M. Isoperimetric functions of groups and exotic cohomology, Combinatorial and geometric group theory (Edinburgh, 1993), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 204 (1995), pp. 87-104

[20] Gromov, M. Hyperbolic groups, Essays in group theory, Springer, New York (Math. Sci. Res. Inst. Publ.) Tome 8 (1987), pp. 75-263 | Article

[21] Gromov, M. Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 182 (1993), pp. 1-295

[22] Heinonen, Juha Lectures on analysis on metric spaces, Springer-Verlag, New York, Universitext (2001), x+140 pages | Article

[23] Mackay, John M. Spaces and groups with conformal dimension greater than one, Duke Math. J., Tome 153 (2010) no. 2, pp. 211-227 | Article

[24] Mackay, John M.; Tyson, Jeremy T. Conformal dimension, American Mathematical Society, Providence, RI, University Lecture Series, Tome 54 (2010), xiv+143 pages (Theory and application) | Article

[25] Maskit, Bernard Kleinian groups, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 287 (1988), xiv+326 pages

[26] Pansu, Pierre Cohomologie L p , espaces homogènes et pincement (Preprint Université Paris-Sud (1999))

[27] Pansu, Pierre Cohomologie L p des variétés à courbure négative, cas du degré 1, Rend. Sem. Mat. Univ. Politec. Torino (1989) no. Special Issue, p. 95-120 (1990) (Conference on Partial Differential Equations and Geometry (Torino, 1988))

[28] Pansu, Pierre Cohomologie L p en degré 1 des espaces homogènes, Potential Anal., Tome 27 (2007) no. 2, pp. 151-165 | Article

[29] Pansu, Pierre Cohomologie L p et pincement, Comment. Math. Helv., Tome 83 (2008) no. 2, pp. 327-357 | Article

[30] Paulin, Frédéric Un groupe hyperbolique est déterminé par son bord, J. London Math. Soc. (2), Tome 54 (1996) no. 1, pp. 50-74 | Article

[31] Rudin, Walter Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg (1973), xiii+397 pages (McGraw-Hill Series in Higher Mathematics)

[32] Whitney, Hassler Geometric integration theory, Princeton University Press, Princeton, N. J. (1957), xv+387 pages