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Normed amenability and bounded cohomology over non-Archimedean fields / Francesco Fournier-Facio

By: Resource type: Ressourcentyp: BuchBookLanguage: English Series: American Mathematical Society. Memoirs of the American Mathematical Society ; volume 299, number 1494 (July 2024)Publisher: Providence : American Mathematical Society, July 2024Description: v, 104 SeitenISBN:
  • 9781470470913
Subject(s): Additional physical formats: Erscheint auch als: Normed Amenability and Bounded Cohomology over Non-Archimedean Fields. Online-Ausgabe 1st ed. Providence : American Mathematical Society, 2024. 1 online resource (116 pages) | Erscheint auch als: Normed amenability and bounded cohomology over non-Archimedean fields. Online-Ausgabe Providence : American Mathematical Society, 2024. 1 Online-Ressource (v, 104 Seiten)MSC: MSC: 22D05 | 20J06 | 55N35 | 22D12 | 22D50Summary: We study continuous bounded cohomology of totally disconnected locally compact groups with coefficients in a non-Archimedean valued field K. To capture the features of classical amenability that induce the vanishing of bounded cohomology with real coefficients, we start by introducing the notion of normed K-amenability, of which we prove an algebraic characterization. It implies that normed K-amenable groups are locally elliptic, and it relates an invariant, the norm of a K-amenable group, to the order of its discrete finite p-subquotients, where p is the characteristic of the residue field of K. Moreover, we prove a characterization of discrete normed K-amenable groups in terms of vanishing of bounded cohomology with coefficients in K. The algebraic characterization shows that normed K-amenability is a very restrictive condition, so the bounded cohomological one suggests that there should be plenty of groups with rich bounded cohomology with trivial K coefficients. We explore this intuition by studying the injectivity and surjectivity of the comparison map, for which surprisingly general statements are available. Among these, we show that if either K has positive characteristic or its residue field has characteristic 0, then the comparison map is injective in all degrees. If K is a finite extension of Qp, we classify unbounded and non-trivial quasimorphisms of a group and relate them to its subgroup structure. For discrete groups, we show that suitable finiteness conditions imply that the comparison map is an isomorphism; this applies in particular to finitely presented groups in degree 2. A motivation as to why the comparison map is often an isomorphism, in stark contrast with the real case, is given by moving to topological spaces. We show that over a non-Archimedean field, bounded cohomology is a cohomology theory in the sense of Eilenberg-Steenrod, except for a weaker version of the additivity axiom which is however equivalent for finite disjoint unions. In particular there exists a Mayer-Vietoris sequence, the main missing piece for computing real bounded cohomology.PPN: PPN: 1902975790
Holdings
Item type Home library Shelving location Call number Status
Institutsbestand Fachbibliothek Mathematik Bibliothek / frei aufgestellt Z 57 c-299.2024,1494 Nicht ausleihbar
Total holds: 0
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