Elements of uniformly bounded word-length in groups

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 $G_{bound}$. 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-length for generating sets of cardinality less than $d$.