Second Bounded Cohomology of a Group: Interpretations Suppose we have a group $\Gamma$ acting on an abelian group $V$. Then it is well-known that the second cohomology group $H^2(\Gamma,V)$ corresponds to equivalence classes of central extensions of $\Gamma$, or equivalently, equivalences classes of short exact sequences of the form
$$1 \longrightarrow V \longrightarrow E \longrightarrow \Gamma \longrightarrow 1$$
This interpretation can also be stated in terms of a lifting property: $H^2(\Gamma,V)$ comprises "obstacles" to the lifting of a homomorphism from $\Gamma$ to a quotient $W/V$ to a homomorphism from $\Gamma$ to $W$, where $W$ is another abelian group containing $V$.

Is there any such natural interpretation of the bounded cohomology group $H_b^2(\Gamma,V)$ in terms of obstacles to lifting?

I know that when $V=\mathbb{R}$ and the action of $\Gamma$ is trivial, then the kernel of the comparison homomorphism 
$$c: H_b^2(\Gamma,\mathbb{R}) \to H^2(\Gamma,\mathbb{R})$$
comprises the space of non-trivial quasi-homomorphisms. 
But what do the elements of the group $H_b^2(\Gamma,V)$ themselves represent for arbitrary $V$? I have been unable to find any satisfactory interpretation in existing literature so far.
 A: Nicolaus Heuer has a paper on the arXiv discussing this.  In it he proves analogous statements of $H^3_{b}$ to the classical interpretation of $H^3$ but for normed $G$-modules.
A: Since you like the correspondence between $H^2(\Gamma,V)$ and central extensions of $\Gamma$, something which has not been mentioned yet which I think you may like is a natural 'geometric' interpretation of $H^2_b(\Gamma,\mathbb{Z})$ when $V=\mathbb{Z}$.
When a central extension of $\Gamma$ by $\mathbb{Z}$ corresponds to a 2-cocycle which is bounded, then the corresponding central extension is in fact quasi-isometric to $\Gamma\times\mathbb{Z}$. This applies to all groups, not just lattices in linear groups. I believe this was first mentioned in "Bounded Cocycles and Combings of Groups" by S.M. Gersten. As such, you may interpret this in terms of obstacles of a more (geo)metric nature (naturally so, since bounded cohomology introduces a norm) and the coarse geometry of the group, which may be why for general $V$ it's harder to say what these precisely are.
