PreprintCC BY 4.0arXiv
Two-generator presentations with a relation gap
Abstract
We exhibit two-generator two-relator group presentations that have cyclic relation modules but do not present one-relator groups; this resolves the relation gap problem. By attaching a single 3-cell to each of the corresponding presentation 2-complexes, one obtains counterexamples to Wall's D2 problem.
Related
- Endomorphic presentations of branch groups
- Binary presentations of center-valued quantification on ortholattices
- Presentations of skein algebras of genus 2
- Computable presentations of reduced group C*-algebras of hyperbolic groups
- Computable presentations of reduced group C*-algebras of groups with the rapid decay property
BibTeX
@misc{bartholdi2026twogenerator,
title = {{Two-generator presentations with a relation gap}},
author = {Laurent Bartholdi and Martin R. Bridson and Roman Mikhailov and John Nicholson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23637},
eprint = {2610.23637},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23637},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Two-generator presentations with a relation gap
AU - Laurent Bartholdi
AU - Martin R. Bridson
AU - Roman Mikhailov
AU - John Nicholson
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23637
UR - https://arxiv.org/abs/2610.23637
AB - We exhibit two-generator two-relator group presentations that have cyclic relation modules but do not present one-relator groups; this resolves the relation gap problem. By attaching a single 3-cell to each of the corresponding presentation 2-complexes, one obtains counterexamples to Wall's D2 problem.
ER -