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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
    Commented Feb 8, 2019 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
    Commented Feb 9, 2019 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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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.

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  • $\begingroup$ Actually it's not only QI to $\Gamma\times\mathbf{Z}$, but QI in a natural way (there's a lift $\Gamma\to\tilde{\Gamma}$ such that $(n,\gamma)\mapsto z^n\tilde{\gamma}$ is a QI). $\endgroup$
    – YCor
    Commented Jul 11, 2019 at 13:20
  • $\begingroup$ I think that the question whether the cyclic center is undistorted implies that the cocycle is bounded (i.e., the cohomology class has a bounded representative) is open, as far as I know. $\endgroup$
    – YCor
    Commented Jul 11, 2019 at 13:21

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