Model Theory and Algorithms for One-Relator Groups
Abstract
We study the elementary theories and algorithmic properties of one-relator groups. We give a first-order sentence characterizing Baumslag-Solitar-freeness in this class. For fixed |m|>1, we regularly interpret BS(1,m) as a definable subgroup in every one-relator group containing it. We uniformly, regularly and injectively interpret integer arithmetic in countable groups of cohomological dimension at most two. We obtain one uniform interpretation for all one-relator groups containing a non-unimodular Baumslag--Solitar subgroup. Every fixed finitely presented group is regularly interpretable in these groups, and their elementary theories are undecidable. We also describe the GBS overgroups of a non-unimodular Baumslag--Solitar subgroup. For the unimodular groups we construct finite-index direct products of free groups and Z, and reduce the elementary theory of BS(k,k) to that of a free group with a named periodic automorphism. Using the recent Open AI result that BS-free one-relator groups are hyperbolic we obtain a terminating algorithm recognizing this class and obtain consequences. From the virtual compact specialness result of another Open AI paper, we obtain an effective construction of virtual free-by-cyclic presentations for hyperbolic one-relator groups, the free kernel may have infinite rank. For classes of torsion-free one-relator groups in which CSA implies toral relative hyperbolicity, we give a uniform algorithm deciding CSA and, in the affirmative case, the existence of a subgroup isomorphic to Z2.
Related
BibTeX
@misc{kharlampovich2026model,
title = {{Model Theory and Algorithms for One-Relator Groups}},
author = {Olga Kharlampovich and Alexei Miasnikov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25305},
eprint = {2610.25305},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25305},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Model Theory and Algorithms for One-Relator Groups
AU - Olga Kharlampovich
AU - Alexei Miasnikov
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25305
UR - https://arxiv.org/abs/2610.25305
AB - We study the elementary theories and algorithmic properties of one-relator groups. We give a first-order sentence characterizing Baumslag-Solitar-freeness in this class. For fixed |m|>1, we regularly interpret BS(1,m) as a definable subgroup in every one-relator group containing it. We uniformly, regularly and injectively interpret integer arithmetic in countable groups of cohomological dimension at most two. We obtain one uniform interpretation for all one-relator groups containing a non-unimodular Baumslag--Solitar subgroup. Every fixed finitely presented group is regularly interpretable in these groups, and their elementary theories are undecidable. We also describe the GBS overgroups of a non-unimodular Baumslag--Solitar subgroup. For the unimodular groups we construct finite-index direct products of free groups and Z, and reduce the elementary theory of BS(k,k) to that of a free group with a named periodic automorphism. Using the recent Open AI result that BS-free one-relator groups are hyperbolic we obtain a terminating algorithm recognizing this class and obtain consequences. From the virtual compact specialness result of another Open AI paper, we obtain an effective construction of virtual free-by-cyclic presentations for hyperbolic one-relator groups, the free kernel may have infinite rank. For classes of torsion-free one-relator groups in which CSA implies toral relative hyperbolicity, we give a uniform algorithm deciding CSA and, in the affirmative case, the existence of a subgroup isomorphic to Z^2.
ER -