Bounded cohomology Hb∗ can
be defined for groups and for topological spaces. Recent work has shown that
Hb∗(M) of a topological space M depends only on Π1(M). In this paper we consider
a new concept—a bounded group—and thereby expand the definition of bounded
cohomology. We prove that bounded cohomology groups are themselves bounded
groups and develop their properties in lower dimensions. In particular, elements of
Hb2(G,A) classify bounded group extensions of G by A. As an application of the
theory of bounded groups we construct the Lyndon spectral sequence. The result
obtained is Theorem 3, which states that Hbn(H,A)G≅bHbn(G,A), when G∕H is
finite.
