of a fixed $\Gamma$-orbit in $X$. The group $\Gamma$ is called *non-elementary* if its limit set contains at least 3 points; then the limit set of $\Gamma$ is in fact uncountable. Using a refinement and an extension of Brooks’ method, Fujiwara showed that for a countable non-elementary group $\Gamma$ of isometries of $X$ acting properly discontinuously on $X$ in a metric sense, the kernel of the map $H_b^2(\Gamma,\mathbb{R})\to H^2(\Gamma,\mathbb{R})$ is infinite dimensional [@F]. Bestvina and Fujiwara extended this result further to countable subgroups of ${\rm Iso}(X)$ whose actions on $X$ satisfy some weaker properness assumption [@BF]. Their result is for example valid for non-elementary subgroups of the *mapping class group* of an oriented surface $S$ of finite type and negative Euler characteristic, i.e. for subgroups of the group of isotopy classes of orientation preserving diffeomorphisms of $S$ which are not virtually abelian. As a consequence, the second bounded cohomology group of every non-elementary subgroup of such a mapping class group is infinite dimensional.

On the other hand, by a result of Burger and Monod [@BM99; @BM02], for every irreducible lattice $\Gamma$ in a connected semi-simple Lie group with finite center, no compact factors and of rank at least 2 the 