PreprintCC BY 4.0arXiv
Simple Torsion-Free Linear Groups from Global Division Algebras
Abstract
We investigate the reduced norm-one group SL1(D) of a central division algebra D of odd degree over ℚ. Using the Brauer-Hasse-Noether theorem and the Margulis-Platonov theorem, we derive a criterion for abstract simplicity in terms of local division completions. For a division algebra of degree $15$ with local invariants 1/3, −1/3, 1/5, and −1/5, the construction yields a candidate example of an infinite, torsion-free, nonabelian simple linear group. This would provide an affirmative answer to Problem 16.78 of the Kourovka Notebook. The argument was generated using an AI system and has not yet been independently verified by human mathematicians.
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BibTeX
@misc{mongo2026simple,
title = {{Simple Torsion-Free Linear Groups from Global Division Algebras}},
author = {Ivan Samuil Nnuga Mongo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37703},
eprint = {2610.37703},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37703},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Simple Torsion-Free Linear Groups from Global Division Algebras
AU - Ivan Samuil Nnuga Mongo
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.37703
UR - https://arxiv.org/abs/2610.37703
AB - We investigate the reduced norm-one group SL_1(D) of a central division algebra D of odd degree over ℚ. Using the Brauer-Hasse-Noether theorem and the Margulis-Platonov theorem, we derive a criterion for abstract simplicity in terms of local division completions. For a division algebra of degree $15$ with local invariants 1/3, −1/3, 1/5, and −1/5, the construction yields a candidate example of an infinite, torsion-free, nonabelian simple linear group. This would provide an affirmative answer to Problem 16.78 of the Kourovka Notebook. The argument was generated using an AI system and has not yet been independently verified by human mathematicians.
ER -