$\Phi$-Harmonic Functions on Discrete Groups and First ${\ell }^{\Phi }$-Cohomology

Abstract

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.

13 pages