The Wiegold problem and free products of left-orderable groups

Abstract

A group has normal rank (or weight) greater than one if no single element normally generates the group. The Wiegold problem from 1976 asks about the existence of a finitely generated perfect group of normal rank greater than one. We show that any free product of nontrivial left-orderable groups has normal rank greater than one. This solves the Wiegold problem by taking free products of finitely generated perfect left-orderable groups, a plethora of which are known to exist. We obtain our estimate of normal rank by a topological argument, proving a type of spectral gap property for an unsigned version of stable commutator length. A key ingredient in the proof is an intricate new construction of a family of left-orders on free products of two left-orderable groups.