On the stabilizers of finite sets of numbers in the R. Thompson group $F$

The subgroups $H_U$ of the R. Thompson group $F$ that are stabilizers of finite sets $U$ of numbers in the interval $(0,1)$ are studied. The algebraic structure of $H_U$ is described and it is proved that the stabilizer $H_U$ is finitely generated if and only if $U$ consists of rational numbers. It is also shown that such subgroups are isomorphic surprisingly often. In particular, if finite sets $U\subset[0,1]$ and $V\subset[0,1]$ consist of rational numbers that are not finite binary fractions, and $|U|=|V|$, then the stabilizers of $U$ and $V$ are isomorphic. In fact these subgroups are conjugate inside a subgroup $\bar F<\operatorname{Homeo}([0,1])$ that is the completion of $F$ with respect to what is called the Hamming metric on $F$. Moreover the conjugator can be found in a certain subgroup $\mathcal F<\bar F$ which consists of possibly infinite tree-diagrams with finitely many infinite branches. It is also shown that the group $\mathcal F$ is non-amenable.