AI 中文总结
本文研究Thompson群$F$的闭有限生成子群的有限性性质，证明了若干等价条件，并推论所有有限生成的极大子群具有$F_\infty$型。
AI 中文摘要
Thompson群 $F$ 的子群 $H$ 是闭的，如果 $F$ 中每个分段-$H$ 函数都属于 $H$。闭子群的典型例子包括无限指数的极大子群、点集的稳定化子和逐点稳定化子，以及Jones的许多子群。一个闭子群是有限生成的当且仅当其Stallings $2$-核是有限的，并且此时它同构于Guba和Sapir意义下关于该核的图群。我们研究 $F$ 的闭有限生成子群在二元有理数上具有有限多个轨道的有限性性质。对于这样的子群 $H$，以下条件等价：$H$ 是 $\mathrm{FP}_2$ 型的；$H$ 是有限表示的；$H$ 是 $F_\infty$ 型的；$H$ 在二元有理数对上具有有限多个轨道；$H$ 在二元有理数上的作用是寡态的。此外，如果 $H$ 在 $(0,1)$ 上的作用是最小的，则这些条件成立当且仅当 $H$ 在 $F$ 的交换化 $\mathbb Z^2$ 中的像的秩为二。所有这些条件都可以从 $H$ 的核判定，并且当它们成立时，可以计算出 $H$ 的有限表示。作为推论，$F$ 的每个有限生成的极大子群都是 $F_\infty$ 型的。
英文摘要
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 $F_\infty$.
