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