Bounded cohomology of discrete groups

TL;DR

This monograph introduces bounded cohomology of discrete groups and topological spaces, covering fundamental definitions, classical results, and applications in geometry and topology, without presenting new original research.

Contribution

It provides a comprehensive, self-contained introduction to bounded cohomology, summarizing key results and applications without new original findings.

Findings

Bounded cohomology helps analyze the simplicial volume of manifolds.

It aids in classifying circle actions and maximal representations.

The theory relates to higher rank flat vector bundles and the Chern conjecture.

Abstract

Bounded cohomology of groups was first defined by Johnson and Trauber during the seventies in the context of Banach algebras. As an independent and very active research field, however, bounded cohomology started to develop in 1982, thanks to the pioneering paper "Volume and Bounded Cohomology" by M. Gromov, where the definition of bounded cohomology was extended to deal also with topological spaces. The aim of this monograph is to provide an introduction to bounded cohomology of discrete groups and of topological spaces. We also describe some applications of the theory to related active research fields (that have been chosen according to the taste and the knowledge of the author). The book is essentially self-contained. Even if a few statements do not appear elsewhere and some proofs are slighlty different from the ones already available in the literature, the monograph does not contain original results. In the first part of the book we settle the fundamental definitions of the theory, and we prove some (by now classical) results on low-dimensional bounded cohomology and on bounded cohomology of topological spaces. Then we describe how bounded cohomology has proved useful in the study of the simplicial volume of manifolds, for the classification of circle actions, for the definition and the description of maximal representations of surface groups, and in the study of higher rank flat vector bundles (also in relation with the Chern conjecture).