Large-scale isoperimetry on locally compact groups and applications

— We introduce various notions of large-scale isoperimetric profile on a locally compact, compactly generated amenable group. These asymptotic quantities provide measurements of the degree of amenability of the group. We are particularly interested in a class of groups with exponential volume growth which are the most amenable possible in that sense. We show that these groups share various interesting properties such as the speed of on-diagonal decay of random walks, the vanishing of the reduced first Lp-cohomology, or the existence of proper isometric actions on Lp whose orbits are almost quasi-isometries. Résumé. — Nous introduisons différentes notions de profil isopérimétrique à grande échelle d’un groupe localement compact, compactement engendré, et moyennable. Ces quantités asymptotiques permettent de mesurer la moyennabilité du groupe. En particulier, nous nous intéressons à la classe des groupes moyennables à croissance exponentielle qui en ce sens sont les “plus moyennables possibles". Nous montrons que ces groupes partagent divers propriétés intéressantes, comme la vitesse de décroissance de la probabilité de retour des marches aléatoires, l’annulation de leur cohomologie-Lp réduite en degré 1, ou bien l’existence d’actions par isométries affines sur Lp dont les orbites sont presque des quasi-isométries. 1. Isoperimetric profiles Let G be a locally compact, compactly generated group equipped with a left-invariant Haar measure μ. Let S be a compact symmetric generating subset of G, i.e. ⋃ n∈N S n = G. Equip G with the left-invariant word metric(1) associated to S, i.e. dS(g, h) = inf{n, g−1h ∈ S}. Let λ be the action of G by left-translations on functions on G, i.e. λ(g)f(x) = f(g−1x). When we restrict to functions in L(G), λ is called the left regular representation of G on L(G). Math. classification: 20-02, 46-02. (1) To have a real metric we must assume that S is symmetric. However, this assumption does not play any role in the sequel