Continuous bounded cohomology of locally compact groups
type: Item
instance of: scholarly article
title: Continuous bounded cohomology of locally compact groups
zbMATH Open document ID: 0967.22006
author: Nicolas Monod
published in: Lecture Notes in Mathematics
publication date: 2001-07-04T00:00:00Z
full work available at URL: http://hdl.handle.net/20.500.11850/145011
full work available at URL: t1758.htm
review text: This volume sets forth a systematic development of continuous bounded cohomology with coefficients in Banach modules. It also gives a number of applications. Chapter one gives an overview of Banach modules and coefficient modules. This includes a discussion of spaces of \(L^\infty\) type as well as remarks on integration. Chapter two studies the notion of relative injectivity. Initially this is carried out in the context of spaces of continuous functions over proper \(G\)-spaces. Then it is set in the context of coefficient modules and related to Zimmer's amenable actions. Chapter three begins the study of continuous bounded cohomology in earnest. The author begins very concretely and then moves on to the functorial approach. Various examples are given as well as the usual functorial properties. Additionally, the relation between continuous bounded cohomology and ordinary continuous cohomology is established via the comparison map. Chapter four is a smattering of techniques that are essential for later applications. Included are discussions of induction, \(L^p\) induction, and ergodicity leading to refined Lyndon-Hochschild-Serre exact sequences. Finally, chapter five presents a number of applications of the theory. For instance, it is related to rough actions, property (TT), quasi-morphisms, actions on the circle, and theorems on irreducible lattices.
reviewed by: Mark R. Sepanski
Mathematics Subject Classification ID: 22E41
Mathematics Subject Classification ID: 22-02
Mathematics Subject Classification ID: 55N35
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 1614276
zbMATH Keywords: continuous bounded cohomology
zbMATH Keywords: Banach modules
zbMATH Keywords: coefficient modules
zbMATH Keywords: \(L^p\) induction
zbMATH Keywords: ergodicity
zbMATH Keywords: Lyndon-Hochschild-Serre exact sequences
zbMATH Keywords: irreducible lattices
MaRDI profile type: Publication
OpenAlex ID: W4205531361
DOI: 10.1007/B80626
description: scientific article; zbMATH DE number 1614276