The two-sided Pompeiu problem for discrete groups

Abstract

We consider a two-sided Pompeiu type problem for a discrete group G G . We give necessary and sufficient conditions for a finite subset K K of G G to have the F ( G ) \mathcal {F}(G) -Pompeiu property. Using group von Neumann algebra techniques, we give necessary and sufficient conditions for G G to be an ℓ 2 ( G ) \ell ^2(G) -Pompeiu group.