Elements of uniformly bounded word-length in groups

2018/11/28 by Amirou, Yanis
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1811.11464

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_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-lengths for generating sets of cardinality less than d.

Related

• On groups with EDT0L word problem
• Short words of infinite order
• Words, permutations, and the nonsolvable length of a finite group
• On profinite groups admitting a word with only few values
• On conciseness of the word in Olshanskii's example