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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.

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    $\begingroup$ Have you found anything in Monod's book? If I recall correctly, some of the non-trivial classes in bounded $H^2$ can have geometric interpretations when one is dealing with e.g. lattices in certain linear groups $\endgroup$ – Yemon Choi Feb 8 at 21:55
  • $\begingroup$ @YemonChoi I have looked at books by Monod and also by Frigerio. There are interpretations by transferring to topological spaces, but I am hoping for a more direct ïnternal"interpretation in terms of lifts say, just like in ordinary cohomology. $\endgroup$ – BharatRam Feb 9 at 13:11
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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.

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