Subgroup of interval exchanges generated by torsion elements and rotations
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- by Michael Boshernitzan
- Proc. Amer. Math. Soc. 144 (2016), 2565-2573
- DOI: https://doi.org/10.1090/proc/12958
- Published electronically: March 1, 2016
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Abstract:
Denote by $G$ the group of interval exchange transformations (IETs) on the unit interval. Let $G_{per}\subset G$ be the subgroup generated by torsion elements in $G$ (periodic IETs), and let $G_{rot}\subset G$ be the subset of $2$-IETs (rotations).
The elements of the subgroup $H=\langle G_{per},G_{rot}\rangle \subset G$ (generated by the sets $G_{per}$ and $G_{rot}$) are characterized constructively in terms of their Sah-Arnoux-Fathi (SAF) invariant. The characterization implies that a non-rotation type $3$-IET lies in $H$ if and only if the lengths of its exchanged intervals are linearly dependent over $\mathbb {Q}$. In particular, $H\subsetneq G$.
The main tools used in the paper are the SAF invariant and a recent result by Y. Vorobets
that $G_{per}$ coincides with the commutator subgroup of $G$.
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Bibliographic Information
- Michael Boshernitzan
- Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005
- MR Author ID: 39965
- Email: michael@rice.edu
- Received by editor(s): November 14, 2013
- Received by editor(s) in revised form: July 26, 2015
- Published electronically: March 1, 2016
- Additional Notes: The author was supported in part by research grant: NSF-DMS-1102298
- Communicated by: Nimish Shah
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2565-2573
- MSC (2010): Primary 37E05, 37E15, 54H15; Secondary 37A05
- DOI: https://doi.org/10.1090/proc/12958
- MathSciNet review: 3477073