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On the stabilizers of finite sets of numbers in the R. Thompson group \(F\). (English) Zbl 1400.20036

St. Petersbg. Math. J. 29, No. 1, 51-79 (2018) and Algebra Anal. 29, No. 1, 70-110 (2017).
Summary: 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 \( \overline F<\operatorname{Homeo}([0,1])\) 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 \( \mathcal {F} < \overline F\) which consists of possibly infinite tree-diagrams with finitely many infinite branches. It is also shown that the group \( \mathcal {F}\) is non-amenable.

MSC:

20F65 Geometric group theory
20G07 Structure theory for linear algebraic groups
20E07 Subgroup theorems; subgroup growth
57M07 Topological methods in group theory

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