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

Abstract

We study the first cohomology groups of a countable discrete group $G$ with coefficients in a $G$-module $\ell^Φ(G)$, where $Φ$ is an $N$-function of class $Δ_2(0)\cap \nabla_2(0)$. In development of ideas of Puls and Martin--Valette, for a finitely generated group $G$, we introduce the discrete $Φ$-Laplacian and prove a theorem on the decomposition of the space of $Φ$-Dirichlet finite functions into the direct sum of the spaces of $Φ$-harmonic functions and $\ell^Φ(G)$ (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 $\overline{H}^{1}(G,\ell^Φ(G)) = 0$. In conclusion, we show the triviality of the first cohomology group for a wreath product of two groups one of which is nonamenable.