showed that for countable groups, vanishing of the second bounded cohomology groups with coefficients in the regular representation is preserved under measure equivalence. Thus second bounded cohomology with coefficients in the regular representation provides an obstruction to the existence of a measure equivalence between two given countable groups.

For the application of these ideas it is necessary to obtain information on these bounded cohomology groups. The first and easiest result in this direction is due to B. Johnson (see [@I] for a discussion and references) who showed that the bounded cohomology of amenable groups with real coefficients is trivial. Later Brooks [@B81] found a combinatorial method for the construction of non-trivial real second bounded cohomology classes and used it to show that the second bounded cohomology group of a finitely generated free group is infinite dimensional.

Fujiwara [@F] investigated the second real bounded cohomology group of a group of isometries of a *hyperbolic* geodesic metric space. Such a space $X$ admits a geometric boundary $\partial X$. Each isometry of $X$ acts as a homeomorphism on $\partial X$. The *limit set* of a group $\Gamma$ of isometries of $X$ is the closed $\Gamma$-invariant subset of $\partial X$ of all accumulation points 