L^\infty(\Gamma,E)^\Gamma \xrightarrow{d} L^\infty
(\Gamma^2,E)^\Gamma\xrightarrow{d} \dots$$ with the usual homogeneous coboundary operator $d$ (see [@M]). There is a natural homomorphism of $H_b^2(\Gamma,E)$ into the ordinary second cohomology group $H^2(\Gamma,E)$ of $\Gamma$ with coefficients $E$ which in general is neither injective nor surjective.

In this paper we are only interested in the case that $E=\mathbb{R}$ with the trivial $\Gamma$-action or that $E=\ell^p(\Gamma)$ for some $p\in (1,\infty)$ with the natural $\Gamma$-action by right translation which assigns to a $p$-summable function $f$ and an element $g\in \Gamma$ the function $gf:h\to f(hg)$.

Since every homomorphism $\rho$ of $\Gamma$ into a countable group $G$ induces a homomorphism $\rho^*:H_b^2(G,\mathbb{R})\to
H_b^2(\Gamma,\mathbb{R})$, second bounded cohomology with real coefficients can be used to find obstructions to the existence of interesting homomorphisms $\Gamma\to G$. The underlying idea is to find conditions on $G$ and $\rho$ which ensure that the image of the map $\rho^*$ is “large” (e.g. infinite dimensional) and conclude that this imposes restrictions on the group $\Gamma$.

Two countable groups $\Gamma,G$ are called *measure equivalent* [@G93] if $\Gamma,G$ admit commuting measure preserving actions on a standard infinite measure Borel space with finite measure fundamental domains. Measure equivalence defines an equivalence relation for countable groups [@Fu99a]. Monod and Shalom [@MS05] 