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Subgroup of interval exchanges generated by torsion elements and rotations


Author: Michael Boshernitzan
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
Published electronically: March 1, 2016
MathSciNet review: 3477073
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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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Additional Information

Michael Boshernitzan
Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005
Email: michael@rice.edu

DOI: https://doi.org/10.1090/proc/12958
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
Article copyright: © Copyright 2016 American Mathematical Society