An obstruction to $\ell ^{p}$-dimension

[Un obstacle à la dimension $\ell ^{p}$]

Monod, Nicolas ; Petersen, Henrik Densing

Annales de l'Institut Fourier, Tome 64 (2014), p. 1363-1371 / Harvested from Numdam

Résumé

Soit $G$ un groupe contenant un sous-groupe infini élémentairement moyennable et soit $2<p<\infty $. Nous construisons des sous-$G$-modules fermés de $\ell ^pG$ d’union croissante dense mais qui rencontrent trivialement un sous-module fermé non trivial. Ce phénomène est un obstacle à la quête d’une dimension $\ell ^p$ et répond à une question de Gaboriau.

Abstract

Let $G$ be any group containing an infinite elementary amenable subgroup and let $2<p<\infty $. We construct an exhaustion of $\ell ^pG$ by closed invariant subspaces which all intersect trivially a fixed non-trivial closed invariant subspace. This is an obstacle to $\ell ^p$-dimension and gives an answer to a question of Gaboriau.

Détail

Publié le : 2014-01-01

MR 3329666 | Zbl 06387310 | Zbl 1309.43001

DOI : https://doi.org/10.5802/aif.2883
Classification:  43A15
Mots clés: dimension $\ell ^p$, analyse harmonique abstraite

Bibliographie

[1] Cheeger, Jeff; Gromov, Mikhael $L_2$-cohomology and group cohomology, Topology, Tome 25 (1986) no. 2, pp. 189-215 | Article | MR 837621 | Zbl 0597.57020

[2] Chou, Ching Elementary amenable groups, Illinois J. Math., Tome 24 (1980) no. 3, pp. 396-407 http://projecteuclid.org/euclid.ijm/1256047608 | MR 573475 | Zbl 0439.20017

[3] Feit, Walter; Thompson, John G. Solvability of groups of odd order, Pacific J. Math., Tome 13 (1963), pp. 775-1029 | Article | MR 166261 | Zbl 0124.26402

[4] Gaboriau, Damien Invariants $l^2$ de relations d’équivalence et de groupes, Publ. Math. Inst. Hautes Études Sci. (2002) no. 95, pp. 93-150 | Article | Numdam | MR 1953191 | Zbl 1022.37002

[5] Gournay, Antoine A dynamical approach to von Neumann dimension, Discrete Contin. Dyn. Syst., Tome 26 (2010) no. 3, pp. 967-987 | Article | MR 2600725 | Zbl 1183.37012

[6] Gournay, Antoine Further properties of $\ell ^p$ dimension, J. Funct. Anal., Tome 266 (2014) no. 2, pp. 487-513 | Article | MR 3132720 | Zbl 1298.43002

[7] Hall, P.; Kulatilaka, C. R. A property of locally finite groups, J. London Math. Soc., Tome 39 (1964), pp. 235-239 | Article | MR 161907 | Zbl 0136.27903

[8] Hayes, Ben An $l^p$-version of von Neumann dimension for Banach space representations of sofic groups II (Preprint, arXiv:1302.2286v2) | Zbl 1300.43002 | Zbl 1339.43001

[9] Hayes, Ben An $l^p$-version of von Neumann dimension for Banach space representations of sofic groups, J. Funct. Anal., Tome 266 (2014) no. 2, pp. 989-1040 | Article | MR 3132735 | Zbl 1300.43002

[10] Hewitt, Edwin; Ross, Kenneth A. Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, Die Grundlehren der mathematischen Wissenschaften, Bd. 115 (1963), pp. viii+519 | MR 156915 | Zbl 0115.10603

[11] Murray, F. J.; Von Neumann, J. On rings of operators, Ann. of Math. (2), Tome 37 (1936) no. 1, pp. 116-229 | Article | MR 1503275 | Zbl 0014.16101

[12] Pansu, Pierre $L^p$-cohomology of symmetric spaces, Geometry, analysis and topology of discrete groups, Int. Press, Somerville, MA (Adv. Lect. Math. (ALM)) Tome 6 (2008), pp. 305-326 | MR 2464400 | Zbl 1159.43003

[13] Puls, Michael J. Zero divisors and $L^p(G)$, Proc. Amer. Math. Soc., Tome 126 (1998) no. 3, pp. 721-728 | Article | MR 1415362 | Zbl 0886.43003

[14] Saeki, Sadahiro On convolution squares of singular measures, Illinois J. Math., Tome 24 (1980) no. 2, pp. 225-232 http://projecteuclid.org/euclid.ijm/1256047718 | MR 575063 | Zbl 0496.42006

[15] Sauer, Roman $L^2$-Betti numbers of discrete measured groupoids, Internat. J. Algebra Comput., Tome 15 (2005) no. 5-6, pp. 1169-1188 | Article | MR 2197826 | Zbl 1099.46045

[16] Tits, J. Free subgroups in linear groups, J. Algebra, Tome 20 (1972), pp. 250-270 | Article | MR 286898 | Zbl 0236.20032