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PreprintCC BY-NC-ND 4.0arXiv

Constructing free subgroups of matrix groups

  • Emmanuel Breuillard
  • Bettina Eick
  • Tommy Hofmann
  • Derek Holt
  • Eamonn O'Brien
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Abstract

The Tits alternative asserts that a finitely generated linear group is either solvable-by-finite or contains a non-abelian free subgroup. Given a non-solvable-by-finite subgroup G of GL(d, F), where F is the field of complex numbers or a non-archimedean local field, we present methods to construct explicitly a non-abelian 2-generator free subgroup of G in specific cases. Our implementations in the computeralgebra system Oscar are publicly available.

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Cite

BibTeXRIS
BibTeX
@misc{breuillard2026constructing,
  title = {{Constructing free subgroups of matrix groups}},
  author = {Emmanuel Breuillard and Bettina Eick and Tommy Hofmann and Derek Holt and Eamonn O'Brien},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03175},
  eprint = {2611.03175},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03175},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Constructing free subgroups of matrix groups
AU  - Emmanuel Breuillard
AU  - Bettina Eick
AU  - Tommy Hofmann
AU  - Derek Holt
AU  - Eamonn O'Brien
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03175
UR  - https://arxiv.org/abs/2611.03175
AB  - The Tits alternative asserts that a finitely generated linear group is either solvable-by-finite or contains a non-abelian free subgroup. Given a non-solvable-by-finite subgroup G of GL(d, F), where F is the field of complex numbers or a non-archimedean local field, we present methods to construct explicitly a non-abelian 2-generator free subgroup of G in specific cases. Our implementations in the computeralgebra system Oscar are publicly available.
ER  -