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 ∆ 2 (0) ∩ ∇ 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 ℓ 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 H 1 (G, ℓ 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.1991 Mathematics Subject Classification. 20J06, 43A07, 43A15.
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