Finiteness properties of closed subgroups of Thompson's group $F$

Abstract

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 $\mathrm{FP}_2$; $H$ is finitely presented; $H$ is of type $F_\infty$; $H$ 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 $\mathbb Z^2$ 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 $F_\infty$.