# 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.

• 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 – Yemon Choi Feb 8 '19 at 21:55
• @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. – BharatRam Feb 9 '19 at 13:11

## 2 Answers

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.

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.

• 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). – YCor Jul 11 '19 at 13:20
• 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. – YCor Jul 11 '19 at 13:21