  Given a family $F$ of subsets of a group $G$ we describe the structure of its
thin-completion $\tau^*(F)$, which is the smallest thin-complete family that
contains $I$. A family $F$ of subsets of $G$ is called thin-complete if each
$F$-thin subset of $G$ belongs to $F$. A subset $A$ of $G$ is called $F$-thin
if for any distinct points $x,y$ of $G$ the intersection $xA\cap yA$ belongs to
the family $F$. We prove that the thin-completion of an ideal in an ideal. If
$G$ is a countable non-torsion group, then the thin-completion $\tau^*(F_G)$ of
the ideal $F_G$ of finite subsets of $G$ is coanalytic but not Borel in the
power-set $P_G$ of $G$.
