Tiling of groups

The following problem formulated by A. M. Vershik connected to several questions in the traectory theory of the finite generated groups pavements is being researched. The result is: let $G$ be decomposed into the free product of two nontrivial groups. Then for any finite subset $S$ of group $G$ there exists a finite subset $P$ of group $G$ including $S$ such that $G$ is being covered by nonintersected left translations of the set $P$.