  We study the first cohomology groups of a countable discrete group $G$ with
coefficients in a $G$-module $\ell^\Phi(G)$, where $\Phi$ is an $N$-function of
class $\Delta_2(0)\cap \nabla_2(0)$. In development of ideas of Puls and
Martin--Valette, for a finitely generated group $G$, we introduce the discrete
$\Phi$-Laplacian and prove a theorem on the decomposition of the space of
$\Phi$-Dirichlet finite functions into the direct sum of the spaces of
$\Phi$-harmonic functions and $\ell^\Phi(G)$ (with an appropriate
factorization). We also prove that if a finitely generated group $G$ has a
finitely generated infinite amenable subgroup with infinite centralizer then
$\overline{H}^{1}(G,\ell^{\Phi}(G)) = 0$. In conclusion, we show the triviality
of the first cohomology group for a wreath product of two groups one of which
is nonamenable.
