{"title":"Relations with a fixed interval exchange transformation","authors":"Magali Jay","doi":"10.1007/s10711-023-00868-x","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>We study the group of all interval exchange transformations (IETs). Katok asked whether it contains a free subgroup. We show that for every IET <em>S</em>, there exists a dense open set <span> <span>\\(\\Omega (S)\\)</span> </span> of admissible IETs such that the group generated by <em>S</em> and any <span> <span>\\(T\\in \\Omega (S)\\)</span> </span> is not free of rank 2. This extends a result by Dahmani et al. (Groups Geom Dyn 7(4):883–910, 2013): the group generated by a generic pair of elements of IET([0;1)) is not free (assuming a suitable condition on the underlying permutation).</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"295 2 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Geometriae Dedicata","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10711-023-00868-x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We study the group of all interval exchange transformations (IETs). Katok asked whether it contains a free subgroup. We show that for every IET S, there exists a dense open set \(\Omega (S)\) of admissible IETs such that the group generated by S and any \(T\in \Omega (S)\) is not free of rank 2. This extends a result by Dahmani et al. (Groups Geom Dyn 7(4):883–910, 2013): the group generated by a generic pair of elements of IET([0;1)) is not free (assuming a suitable condition on the underlying permutation).
期刊介绍：
Geometriae Dedicata concentrates on geometry and its relationship to topology, group theory and the theory of dynamical systems.
Geometriae Dedicata aims to be a vehicle for excellent publications in geometry and related areas. Features of the journal will include:
A fast turn-around time for articles.
Special issues centered on specific topics.
All submitted papers should include some explanation of the context of the main results.
Geometriae Dedicata was founded in 1972 on the initiative of Hans Freudenthal in Utrecht, the Netherlands, who viewed geometry as a method rather than as a field. The present Board of Editors tries to continue in this spirit. The steady growth of the journal since its foundation is witness to the validity of the founder''s vision and to the success of the Editors'' mission.
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