PreprintCC BY 4.0arXiv
Solvable groups defined by redundant cyclic presentations
Abstract
We study the two-generator one-relator groups defined by ⟨ x0,x1| U(x0,x1)U(x1,x0)−1⟩, and determine exactly which of them fail to contain a free subgroup of rank $2$. We show that every such group is a solvable Baumslag--Solitar group BS(1,m), and that every integer value m can occur. This disproves a conjecture predicting that only m∈ {−1,0,1} can occur.
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BibTeX
@misc{chinyere2026solvable,
title = {{Solvable groups defined by redundant cyclic presentations}},
author = {Ihechukwu Chinyere},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08863},
eprint = {2609.08863},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08863},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Solvable groups defined by redundant cyclic presentations
AU - Ihechukwu Chinyere
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08863
UR - https://arxiv.org/abs/2609.08863
AB - We study the two-generator one-relator groups defined by ⟨ x_0,x_1| U(x_0,x_1)U(x_1,x_0)^−1⟩, and determine exactly which of them fail to contain a free subgroup of rank $2$. We show that every such group is a solvable Baumslag--Solitar group BS(1,m), and that every integer value m can occur. This disproves a conjecture predicting that only m∈ {−1,0,1} can occur.
ER -