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.

  • 1
    $\begingroup$ 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

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.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.