Soficity of One-Relator Groups and Reducible Presentations

Abstract

We prove that every one-relator group is sofic, answering a question of Nate Brown. More generally, every group admitting a reducible presentation without proper powers is sofic, with one proper-power relation allowed at the final reducible step. The proof starts from the endpoint-preserving edge replacements of Poulin--Wróbel. We retain the free-group word carried by each terminal replacement and use it to define an F(S)-valued cocycle repair; in its relative form, the repair preserves the previously chosen generator coordinates while controlling the new relator defect. Coinduction transports the lower cocycle through successive one-relator-product extensions, allowing the construction to be iterated along reducible presentations. For a final relator w^m, cyclic translates of the repaired set amplify a zero-defect set of measure close to 1/m to one of measure close to one. A cocycle criterion then converts arbitrarily small relator defect into finite permutation approximations via a treeable skew-product orbit relation.

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BibTeX RIS

BibTeX

@misc{son2026soficity,
  title = {{Soficity of One-Relator Groups and Reducible Presentations}},
  author = {Jinhyuk Son},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02213},
  eprint = {2610.02213},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02213},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Soficity of One-Relator Groups and Reducible Presentations
AU  - Jinhyuk Son
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.02213
UR  - https://arxiv.org/abs/2610.02213
AB  - We prove that every one-relator group is sofic, answering a question of Nate Brown. More generally, every group admitting a reducible presentation without proper powers is sofic, with one proper-power relation allowed at the final reducible step. The proof starts from the endpoint-preserving edge replacements of Poulin--Wróbel. We retain the free-group word carried by each terminal replacement and use it to define an F(S)-valued cocycle repair; in its relative form, the repair preserves the previously chosen generator coordinates while controlling the new relator defect. Coinduction transports the lower cocycle through successive one-relator-product extensions, allowing the construction to be iterated along reducible presentations. For a final relator w^m, cyclic translates of the repaired set amplify a zero-defect set of measure close to 1/m to one of measure close to one. A cocycle criterion then converts arbitrarily small relator defect into finite permutation approximations via a treeable skew-product orbit relation.
ER  -

Abstract

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