Orlicz Spaces and the First Orlicz Cohomology of Discrete Groups

We study Orlicz spaces on discrete groups. First we consider the first Orlicz cohomology of general (not necessarily countable) discrete groups. We give some conditions for the triviality of the first Orlicz cohomology and the first reduced Orlicz cohomology of a discrete group and for the coincidence of these spaces.
Then we deal with a group-theoretic version of the so-called Pompeiu problem. We introduce the notion of a $\Phi$-zero divisor, where $\Phi$ is a Young function, on a discrete group. We extend some results about $p$-divisors on free groups by Linnell and Puls to $\Phi$-zero divisors.