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We study the first cohomology groups of a countable discrete group $G$ withcoefficients in a $G$-module $\ell^\Phi(G)$, where $\Phi$ is an $N$-function ofclass $\Delta_2(0)\cap \nabla_2(0)$. In development of ideas of Puls andMartin--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 appropriatefactorization). We also prove that if a finitely generated group $G$ has afinitely generated infinite amenable subgroup with infinite centralizer then$\overline{H}^{1}(G,\ell^{\Phi}(G)) = 0$. In conclusion, we show the trivialityof the first cohomology group for a wreath product of two groups one of whichis nonamenable.

$\Phi$-Harmonic Functions on Discrete Groups and First $\ell^\Phi$-Cohomology