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(bound). We give sufficient criteria for triviality and finiteness of G(bound). We prove that if G is virtually abelian then Gbound 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.