Some results on bounded cohomology

Summary

Abstract

The structure of the second bounded cohomology group is investigated. This group is computed for a free group, a torus knot group and a surface group. The description is based on the notion of a pseudocharacter. A survey of results on bounded cohomology is given.

Introduction

If we use the standard bar resolution then the definition of bounded cohomology of the trivial G-module ℝ differs from the definition of ordinary cohomology in that instead of arbitrary cochains with values in ℝ one should consider only the bounded cochains.

Bounded cohomology was first defined for discrete groups by F. Trauber and then for topological spaces by M.Gromov [39]. Moreover, M.Gromov developed the theory of bounded cohomology and applied it to Riemannian geometry, thus demonstrating the importance of this theory. The second bounded cohomology group is related to some topics of the theory of right orderable groups and has applications in the theory of groups acting on a circle [33], [55], [56].

In [9] R.Brooks made a first step in understanding the theory of bounded cohomology from the point of view of relative homological algebra. This approach was developed by N.Ivanov [48], whose paper probably contains the best introduction in the subject.

Actually the theory of bounded cohomology of discrete groups is a part of the theory of cohomology in topological groups [34] and in Banach algebras [49] introduced at the beginning of the sixties if we consider the trivial (that is g_x = x = xg) l₁(G)-module ℝ.