On the stabilizers of finite sets of numbers in the R. Thompson group $F$
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- by G. Golan and M. Sapir
- St. Petersburg Math. J. 29 (2018), 51-79
- 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 $|U|=|V|$, 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.References
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Bibliographic Information
- G. Golan
- Affiliation: Vanderbilt University, 2201 West End Ave, Nashville, TN 37235, USA
- MR Author ID: 1031418
- ORCID: 0000-0002-9868-8274
- Email: gilgula.g@gmail.com, gili.golan@vanderbilt.edu
- M. Sapir
- Affiliation: Vanderbilt University, 2201 West End Ave, Nashville, TN 37235, USA
- MR Author ID: 189574
- Email: markvs@gmail.com, m.sapir@vanderbilt.edu
- 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.
- © Copyright 2017 American Mathematical Society
- Journal: St. Petersburg Math. J. 29 (2018), 51-79
- MSC (2010): Primary 20F65, 20G07
- DOI: https://doi.org/10.1090/spmj/1482
- MathSciNet review: 3660685
Dedicated: Dedicated to Professor Yuri Burago on the occasion of his 80th birthday