The Wiegold problem and free products of left-orderable groups

TL;DR

This paper proves that free products of nontrivial left-orderable groups have normal rank greater than one, solving the Wiegold problem by combining topological methods and new constructions of left-orders.

Contribution

It demonstrates that free products of nontrivial left-orderable groups have normal rank greater than one, addressing a longstanding open problem.

Findings

Any free product of nontrivial left-orderable groups has normal rank > 1.

Constructs a new family of left-orders on free products of two groups.

Establishes a spectral gap property for an unsigned version of stable commutator length.

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.