$\Phi$-Harmonic Functions on Discrete Groups and 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 $\Delta_2(0)\cap \nabla_2(0)$. 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.

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