  We study a characteristic subgroup of finitely generated groups, consisting
of elements with uniform upper bound for word-lengths. For a group $G$, we
denote this subgroup by $G_{bound}$. We give sufficient criteria for triviality
and finiteness of $G_{bound}$. We prove that if $G$ is virtually abelian then
$G_{bound}$ is finite. In contrast with numerous examples where $G_{bound}$ is
trivial, we show that for every finite group $A$, there exists an infinite
group $G$ with $G_{bound}=A$. This group $G$ can be chosen among torsion
groups. We also study the group $G_{bound}(d)$ of elements with uniformly
bounded word-lengths for generating sets of cardinality less than $d$.
