An obstruction to p -dimension
[Un obstacle à la dimension p ]
Annales de l'Institut Fourier, Tome 64 (2014) no. 4, pp. 1363-1371.

Soit G un groupe contenant un sous-groupe infini élémentairement moyennable et soit 2<p<. Nous construisons des sous-G-modules fermés de p G 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  p et répond à une question de Gaboriau.

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

DOI : 10.5802/aif.2883
Classification : 43A15
Keywords: $\ell ^p$-dimension, abstract harmonic analysis
Mot clés : dimension $\ell ^p$, analyse harmonique abstraite
Monod, Nicolas 1 ; Petersen, Henrik Densing 2

1 École Polytechnique Fédérale de Lausanne Station 8, CH-1015 Lausanne (Switzerland)
2 University of Copenhagen Department of Mathematical Sciences Universitetsparken 5 2100 København Ø(Denmark)
@article{AIF_2014__64_4_1363_0,
     author = {Monod, Nicolas and Petersen, Henrik Densing},
     title = {An obstruction to $\ell ^{p}$-dimension},
     journal = {Annales de l'Institut Fourier},
     pages = {1363--1371},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
     number = {4},
     year = {2014},
     doi = {10.5802/aif.2883},
     mrnumber = {3329666},
     zbl = {1309.43001},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2883/}
}
TY  - JOUR
AU  - Monod, Nicolas
AU  - Petersen, Henrik Densing
TI  - An obstruction to $\ell ^{p}$-dimension
JO  - Annales de l'Institut Fourier
PY  - 2014
SP  - 1363
EP  - 1371
VL  - 64
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2883/
DO  - 10.5802/aif.2883
LA  - en
ID  - AIF_2014__64_4_1363_0
ER  - 
%0 Journal Article
%A Monod, Nicolas
%A Petersen, Henrik Densing
%T An obstruction to $\ell ^{p}$-dimension
%J Annales de l'Institut Fourier
%D 2014
%P 1363-1371
%V 64
%N 4
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2883/
%R 10.5802/aif.2883
%G en
%F AIF_2014__64_4_1363_0
Monod, Nicolas; Petersen, Henrik Densing. An obstruction to $\ell ^{p}$-dimension. Annales de l'Institut Fourier, Tome 64 (2014) no. 4, pp. 1363-1371. doi : 10.5802/aif.2883. http://archive.numdam.org/articles/10.5802/aif.2883/

[1] Cheeger, Jeff; Gromov, Mikhael L 2 -cohomology and group cohomology, Topology, Volume 25 (1986) no. 2, pp. 189-215 | DOI | MR | Zbl

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

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

[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 | DOI | EuDML | Numdam | MR | Zbl

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

[6] Gournay, Antoine Further properties of p dimension, J. Funct. Anal., Volume 266 (2014) no. 2, pp. 487-513 | DOI | MR | Zbl

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

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

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

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

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

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

[13] Puls, Michael J. Zero divisors and L p (G), Proc. Amer. Math. Soc., Volume 126 (1998) no. 3, pp. 721-728 | DOI | MR | Zbl

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

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

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

Cité par Sources :