Finiteness properties of closed subgroups of Thompson's group F

@misc{golan2026finiteness,
  title = {{Finiteness properties of closed subgroups of Thompson's group $F$}},
  author = {Gili Golan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14702},
  eprint = {2609.14702},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14702},
}

TY  - UNPB
TI  - Finiteness properties of closed subgroups of Thompson's group F
AU  - Gili Golan
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14702
UR  - https://arxiv.org/abs/2609.14702
AB  - A subgroup H of Thompson's group F is closed if every piecewise-H function in F belongs to H. Prominent examples of closed subgroups are the maximal subgroups of infinite index, stabilizers and pointwise stabilizers of sets of points, and many of Jones' subgroups. A closed subgroup is finitely generated if and only if its Stallings $2$-core is finite, and it is then isomorphic to a diagram group over the core, in the sense of Guba and Sapir. We study the finiteness properties of closed finitely generated subgroups of F with finitely many orbits on the dyadic rationals. For such a subgroup H the following are equivalent: H is of type FP_2; H is finitely presented; H is of type F_∞; H has finitely many orbits on pairs of dyadic rationals; the action of H on the dyadic rationals is oligomorphic. If moreover the action of H on (0,1) is minimal, these conditions hold if and only if the image of H in the abelianization ℤ^2 of F has rank two. All these conditions can be decided from the core of H, and when they hold a finite presentation of H can be computed. As a consequence, every finitely generated maximal subgroup of F is of type F_∞.
ER  -