Second cohomology of a torsion-free hyperbolic group

Question

Does any body know examples of torsion-free hyperbolic group $G$ such that $H^2(G,\mathbb{R})=0$ (trivial G-action on $\mathbb{R}$)?

In fact I am interested in if there are known examples of even-dimensional, closed Riemannian manifolds $(M,g)$ with negative sectional curvature such that $b_2(M)=0$.

I noticed this paper by Epstein-Fujiwara, but I am not sure if the result for bounded group cohomology gives information on my question.

Answer 1

Based on my comments:

First of all, the question about hyperbolic groups is very different from the one about fundamental groups of closed connected manifolds of negative curvature: "Most" hyperbolic groups are very much unlike "manifold groups." Second: The mentioned paper by Epstein and Fujiwara is interesting but totally irrelevant for the purpose of your question. Now, your real question is:

Is there an example of a closed connected even-dimensional manifold $M$ of negative curvature such that $b_2(M)=0$?

Here is what I know: The first interesting case, of course, is of 4-dimensional manifolds. Such a manifold $M$ would have positive Euler characteristic (see the references here), hence, effectively, you are asking about the existence of a negatively curved 4-dimensional rational homology sphere. This is an open problem (stated explicitly for manifolds of constant negative curvature by Bruno Martelli, I think). If there is such a hyperbolic 4-manifold, it would have the smallest possible volume among hyperbolic 4-manifolds.

Among locally-symmetric manifolds of negative curvature, complex-hyperbolic ones always have $b_2>0$ (because of the Kahler class). I do not believe there are any explicitly known examples (say, meaning that somebody computed their Betti numbers) of closed real-hyperbolic manifolds of dimension $\ge 6$. There are also no known vanishing theorems for $b_2$ in the class of manifolds. (All the known results are on the "nonvanishing side", they are of the type: There exists a finite-sheeted covering space with positive Betti numbers $b_i$ so some values of $i$.) This leaves one with quotients of quaternionic-hyperbolic spaces (and of the Cayley-hyperbolic plane). While there are no explicitly known examples (again, meaning that somebody computed Betti numbers), there might be vanishing/nonvanishing theorems for $b_2$ known in this class.

As for negatively curved manifolds of dimension $\ge 4$ which are not locally-symmetric, there is only a handful of constructions (which mostly use locally-symmetric manifolds as their starting point) and no known construction can ensure vanishing of $b_2$.

Thus, unless there are known vanishing results for $b_2$ in the case of torsion-free cocompact discrete subgroups of isometries of quaternionic-hyperbolic spaces ${\mathbf H}{\mathbb H}^n, n\ge 2$, your question should be treated as an open problem.