The Pompeiu Problem and Discrete Groups
arXiv:1202.2398
Abstract
We formulate a version of the Pompeiu problem in the discrete group setting. Necessary and sufficient conditions are given for a finite collection of finite subsets of a discrete abelian group, whose torsion free rank is less than the cardinal of the continuum, to have the Pompeiu property. We also prove a similar result for nonabelian free groups. A sufficient condition is given that guarantees the harmonicity of a function on a nonabelian free group if it satisfies the mean-value property over two spheres.
Version two fixes some typos. Also a new section was added concerning the harmonicity of a function with regards to the mean-value property on two spheres. Corrected some more typos