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


Authors: G. Golan and M. Sapir
Original publication: Algebra i Analiz, tom 29 (2017), nomer 1.
Journal: St. Petersburg Math. J. 29 (2018), 51-79
MSC (2010): Primary 20F65, 20G07
DOI: https://doi.org/10.1090/spmj/1482
Published electronically: December 27, 2017
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Abstract: 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 $ \vert U\vert=\vert V\vert$, then the stabilizers of $ U$ and $ V$ are isomorphic. In fact these subgroups are conjugate inside a subgroup $ \overline F<\mathrm {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.


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Additional Information

G. Golan
Affiliation: Vanderbilt University 2201 West End Ave Nashville, TN 37235 USA
Email: gilgula.g@gmail.com\\ gili.golan@vanderbilt.edu

M. Sapir
Affiliation: Vanderbilt University 2201 West End Ave Nashville, TN 37235 USA
Email: markvs@gmail.com\\ m.sapir@vanderbilt.edu

DOI: https://doi.org/10.1090/spmj/1482
Keywords: Thompson group $F$, stabilizers
Received by editor(s): May 15, 2016
Published electronically: December 27, 2017
Additional Notes: The research of the first author was supported in part by a Fulbright grant and a post-doctoral scholarship of Bar-Ilan University, the research of the second author was supported in part by the NSF grants DMS 1418506, DMS 1318716.
Dedicated: Dedicated to Professor Yuri Burago on the occasion of his 80th birthday
Article copyright: © Copyright 2017 American Mathematical Society

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