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Centralizers in the group of interval exchange transformations

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Abstract

We study the group of interval exchange transformations. Let T be an m-interval exchange transformation. By the rank of T we mean the dimension of the ℚ-vector space spanned by the lengths of the exchanged subintervals. We prove that if T satisfies Keane’s infinite distinct orbit condition and rank(T) > 1 + [m/2], then the only interval exchange transformations which commute with T are its powers.

In the case that T is a minimal 3-interval exchange transformation, we prove a more precise result: T has a trivial centralizer in the group of interval exchange transformations if and only if T satisfies the infinite distinct orbit condition.

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Acknowledgments

I would like to thank Michael Boshernitzan for encouraging me to investigate this topic, for reading many drafts of this paper, and for indicating how to prove Proposition 6.3. I would also like to thank the referee for reading the paper carefully and making several helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Rice University, Houston, TX, 77005, USA

    Daniel Bernazzani

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  1. Daniel Bernazzani
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Correspondence to Daniel Bernazzani.

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Bernazzani, D. Centralizers in the group of interval exchange transformations. Isr. J. Math. 233, 29–48 (2019). https://doi.org/10.1007/s11856-019-1902-6

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  • DOI: https://doi.org/10.1007/s11856-019-1902-6

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