Large-scale isoperimetry on locally compact groups and applications. (English) Zbl 1167.22004

Actes de Séminaire de Théorie Spectrale et Géométrie. Année 2006–2007. St. Martin d’Hères: Université de Grenoble I, Institut Fourier. Séminaire de Théorie Spectrale et Géométrie 25, 179-188 (2007).
The author puts together results from several of his recent papers [Ann. Inst. Fourier 59, No. 2, 851–876 (2009; Zbl 1225.22019)], [Rev. Mat. Iberoam. 24, No. 3, 825–864 (2008; Zbl 1194.53036)], [Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces, to appear in Comment. Math. Helv.] and [Isoperimetric profile of subgroups and probability of return of random walks on elementary solvable groups, per bibl.]. The common idea under all these results is to show that groups in the class of so-called geometrically elementary solvable groups give optimal results, among groups with exponential volume growth, when several geometric indicators of the degree of amenability are applied to them. The class of geometrically elementary solvable groups is a class of locally compact groups invariant under finite products, quotients, unimodular closed compactly generated subgroups and quasi-isometric images, built from linear solvable groups but that contains some non-solvable groups. No proofs are given.
For the entire collection see [Zbl 1144.35003].


22D10 Unitary representations of locally compact groups
22E40 Discrete subgroups of Lie groups
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
Full Text: EuDML