FREE GROUPS IN THE SECOND BOUNDED COHOMOLOGY

The second bounded cohomology of a free group of rank greater than 1 is infinite dimensional as a vector space over R [ 4]. For a group G and its nth commutator subgroup G((n)), the quotient G/G((n)) is amenable and the homomorphism (H) over cap (2)(G) --> (H) over cap (2)(G((n))) induced from the inclusion homomorphism G((n)) --> G is injective.In this article, we prove that if G((n)) is free of rank greater than 1 for some finite ordinal n, then G is residually solvable and its second bounded cohomology is infinite dimensional. We prove its converse for a group generated by two elements. As for groups that are not residually solvable, we investigate the dimension of the second bounded cohomology of a perfect group. Also, some results on bounded cohomology of a connected CW complex X by applying a Quillen's plus construction X+ to kill a perfect normal subgroup of pi X-1 are given.