Elements of uniformly bounded word-length in groups

Abstract

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 $d$.