PreprintCC BY 4.0arXiv
Invariant random subgroups of the finitary special linear group over a finite field
Abstract
An invariant random subgroup of a discrete group is a conjugation-invariant probability measure on the space of its subgroups. We confirm a conjecture of Simon Thomas and classify the invariant random subgroups of the finitary special and general linear groups over a finite field. These are shown to be stabilizers of random tuples of vectors and functionals. The classification uses the character theory of the finitary special linear group established by Skudlarek and the theory of the invariant random subgroups of the finitary permutation group established by Vershik. In addition, we classify the invariant probability measures for several natural actions of these finitary groups.
Related
- Invariant random subgroups of C*-simple groups
- Complete mappings stabilizing Sylow $2$-subgroups in central quotients of special linear groups
- Abelian quotients and orbit sizes for subgroups of linear groups
- Möbius functions for pseudo-Levi subgroups in finite general linear and symplectic groups
- Metrizable and Fréchet--Urysohn subgroups and linear subspaces of topological vector spaces
BibTeX
@misc{atar2026invariant,
title = {{Invariant random subgroups of the finitary special linear group over a finite field}},
author = {Noam Atar and Arie Levit},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03960},
eprint = {2611.03960},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03960},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Invariant random subgroups of the finitary special linear group over a finite field
AU - Noam Atar
AU - Arie Levit
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03960
UR - https://arxiv.org/abs/2611.03960
AB - An invariant random subgroup of a discrete group is a conjugation-invariant probability measure on the space of its subgroups. We confirm a conjecture of Simon Thomas and classify the invariant random subgroups of the finitary special and general linear groups over a finite field. These are shown to be stabilizers of random tuples of vectors and functionals. The classification uses the character theory of the finitary special linear group established by Skudlarek and the theory of the invariant random subgroups of the finitary permutation group established by Vershik. In addition, we classify the invariant probability measures for several natural actions of these finitary groups.
ER -