Introduction

Abstract

In his celebrated 1982 I.H.É.S. paper Volume and bounded cohomology [69], M. Gromov considers the singular bounded cohomology H^• _b(M) of a manifold M and proves a spray of deep and surprising theorems on the geometry of manifolds. This cohomology H^• _b(M) is defined exactly like usual singular cohomology, except that all cochains are required to be bounded. Similarly, one can define for a group Г the bounded cohomology H^• _b(Г) by the usual inhomogeneous complex with the only change that one restricts attention to bounded cochains : $O \to R \to \ell ^\infty \left( {\Gamma ,R} \right) \to \ell ^\infty \left( {\Gamma ^2 ,R} \right) \to \bullet \bullet \bullet$