Geodesic
On the stabilizers of finite sets of numbers in the R.~Thompson group~$F$
Algebra i analiz, Tome 29 (2017) no. 1, pp. 70-110.

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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.
Keywords: Thompson group $F$, stabilizers. 
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G. Golan; M. Sapir. On the stabilizers of finite sets of numbers in the R.~Thompson group~$F$. Algebra i analiz, Tome 29 (2017) no. 1, pp. 70-110. http://geodesic.mathdoc.fr/item/AA_2017_29_1_a4/

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