Elements of uniformly bounded word-length in groups (Q2077235)

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