The Wiegold problem and free products of left-orderable groups
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914176890306560 |
|---|---|
| author | Chen, Lvzhou Lodha, Yash |
| author_facet | Chen, Lvzhou Lodha, Yash |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26073 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Wiegold problem and free products of left-orderable groups Chen, Lvzhou Lodha, Yash Group Theory Geometric Topology 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. |
| title | The Wiegold problem and free products of left-orderable groups |
| topic | Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2510.26073 |