geodesic $\gamma$ connecting $x$ to $y$ meets the $m$-neighborhood of $x_0$ and if $d(x,y)\geq R(x_0,m)$ then there are at most $c(x_0,m)$ elements $g\in \Gamma$ such that $d(x,gx)\leq m$ and $d(y,gy)\leq
m$ (compare with the definition of an acylindrical isometry group in [@B03]). We show in Section 4 (see [@F; @BF; @MMS] for earlier results).

[**Theorem A:**]{} [*Let $\Gamma$ be a non-elementary weakly acylindrical countable group of isometries of an arbitrary hyperbolic geodesic metric space. Then the kernels of the maps $H_b^2(\Gamma,\mathbb{R})\to H^2(\Gamma,\mathbb{R})$ and $H_b^2(\Gamma,\ell^p(\Gamma))\to H^2(\Gamma,\ell^p(\Gamma))$ $
(1< p<\infty)$ are infinite dimensional.*]{}

As an easy corollary of Theorem A and a result of Bowditch [@B03] we obtain an extension of the result of Bestvina and Fujiwara [@BF]. For its formulation, we say that a group $\Gamma$ *virtually splits* as a direct product if $\Gamma$ has a finite index subgroup $\Gamma^\prime$ which splits as a direct product of two infinite groups. We show.

The following corollary is an immediate consequence of Corollary B and the work of Burger-Monod and Monod-Shalom [@BM99; @BM02; @MS05]. For its formulation, call a lattice $\Lambda$ in a product $G=G_1\times G_2$ of two locally compact $\sigma$-compact and non-compact topological groups *irreducible* if the projection of $\Lambda$ to each 