Maximal subgroups of topological full groups arising from factor maps and subgroupoids
Abstract
We construct maximal subgroups of topological full groups and their commutator subgroups using factor maps and wide open subgroupoids of minimal effective ample groupoids with Cantor unit spaces, assuming almost finiteness or pure infiniteness. For two-to-one coverings and prime factor maps with principal target and fibers of cardinality at most two, surjectivity and injectivity of the same induced map on full-group abelianizations characterize maximality of the factor full group and factor commutator subgroup, respectively. For subgroupoids arising from prime-order cyclic actions free on the unit space, the normalizer of the smaller commutator subgroup in the ambient full group is a semidirect product, maximal exactly when the inclusion induces a surjection on full-group abelianizations. In both constructions, the embedded commutator subgroup has a unique maximal overgroup in the ambient commutator subgroup, namely its normalizer. For Cantor minimal systems, the factor construction yields partition stabilizers of the type in the Grigorchuk-Vorobets conjecture. The subgroupoid normalizers act topologically primitively. Groupoid homology makes the criteria computable for these systems and shifts of finite type. Examples include finitely generated nonsimple and locally finite simple maximal subgroups of infinite index in finitely generated infinite simple amenable groups.
Related
- Stabilizers and maximal subgroups of topological full groups
- Maximal subgroups of topological full groups of shifts of finite type
- A note on maximal subgroups of countable groups
- On second maximal subgroups of finite groups
- Maximal ranks of free abelian subgroups of generalized cactus groups associated with finite Coxeter systems
BibTeX
@misc{matui2026maximal,
title = {{Maximal subgroups of topological full groups arising from factor maps and subgroupoids}},
author = {Hiroki Matui},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.33118},
eprint = {2609.33118},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33118},
}
RIS
TY - UNPB
TI - Maximal subgroups of topological full groups arising from factor maps and subgroupoids
AU - Hiroki Matui
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2609.33118
UR - https://arxiv.org/abs/2609.33118
AB - We construct maximal subgroups of topological full groups and their commutator subgroups using factor maps and wide open subgroupoids of minimal effective ample groupoids with Cantor unit spaces, assuming almost finiteness or pure infiniteness. For two-to-one coverings and prime factor maps with principal target and fibers of cardinality at most two, surjectivity and injectivity of the same induced map on full-group abelianizations characterize maximality of the factor full group and factor commutator subgroup, respectively. For subgroupoids arising from prime-order cyclic actions free on the unit space, the normalizer of the smaller commutator subgroup in the ambient full group is a semidirect product, maximal exactly when the inclusion induces a surjection on full-group abelianizations. In both constructions, the embedded commutator subgroup has a unique maximal overgroup in the ambient commutator subgroup, namely its normalizer. For Cantor minimal systems, the factor construction yields partition stabilizers of the type in the Grigorchuk-Vorobets conjecture. The subgroupoid normalizers act topologically primitively. Groupoid homology makes the criteria computable for these systems and shifts of finite type. Examples include finitely generated nonsimple and locally finite simple maximal subgroups of infinite index in finitely generated infinite simple amenable groups.
ER -