Φ-Harmonic functions on discrete groups and the first ℓ^Φ-cohomology

We study the first cohomology groups of a countable discrete group G with coefficients in a G-module ℓ^Φ(G), where Φ is an N-function of class Δ₂(0) ∩ ▿₂(0). Developing the 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 ℓ^Φ(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 or (Formula presented.). In conclusion, we show the triviality of the first cohomology group for the wreath product of two groups one of which is nonamenable.