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

Monod, Nicolas; Petersen, Henrik Densing

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 $2<p<\infty$. Nous construisons des sous-$G$-modules fermés de ${\ell }^{p}G$ 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 ${\ell }^p$ et répond à une question de Gaboriau.

Let $G$ be any group containing an infinite elementary amenable subgroup and let $2<p<\infty$. We construct an exhaustion of ${\ell }^{p}G$ 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.

Classification : 43A15
Keywords: $\ell ^p$-dimension, abstract harmonic analysis
Mot clés : dimension $\ell ^p$, analyse harmonique abstraite

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