Generating groups by conjugation-invariant sets

TL;DR

This paper introduces the concept of finite C-width in groups, explores examples including Thompson's group F, and studies how this property behaves under group-theoretic constructions.

Contribution

It defines finite C-width, provides examples including Thompson's group F, and analyzes the stability of this property under group extensions.

Findings

Thompson's group F's commutator subgroup has finite C-width.

The class of groups with finite C-width is closed under group extensions.

Multiple examples of groups with finite C-width are given.

Abstract

Let S be a generating set of a group G. We say that G has FINITE WIDTH relative to S if G=(S\cup S^{-1})^k for a suitable natural number k. We say that a group G is a group of FINITE C-WIDTH if G has finite width with respect to all conjugation-invariant generating sets. We give a number of examples of groups of finite C-width, and, in particular, we prove that the commutator subgroup F' of Thompson's group F is a group of finite C-width. We also study the behaviour of the class of all groups of finite C-width under some group-theoretic constructions; it is established, for instance, that this class is closed under formation of group extensions.