$\Phi$-harmonic functions on discrete groups and the first $\ell^\Phi$-cohomology

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 $\Delta_2(0)\cap\nabla_2(0)$. Developing the 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 prove also 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 the wreath product of two groups one of which is nonamenable.