no. 4
Uniform perfectness for Interval Exchange Transformations with or without Flips
[Perfection uniforme pour les échanges d’intervalles avec ou sans Flips]
Guelman, Nancy1 ; Liousse, Isabelle2
1 IMERL, Facultad de Ingeniería Universidad de la República C.C. 30, Montevideo (Uruguay)
2 Univ. Lille, CNRS UMR 8524 - Laboratoire Paul Painlevé F-59000 Lille (France)
Annales de l'Institut Fourier, Tome 72 (2022) no. 4, pp. 1477-1501.

Soit 𝒢 le groupe des échanges d’intervalles. Des résultats d’Arnoux–Fathi, Sah et Vorobets indiquent que 𝒢 0 le sous-groupe de 𝒢 engendré par ses commutateurs est simple. Arnoux prouve que le groupe 𝒢 ¯ des échanges d’intervalles avec flips est simple.

Nous établissons que tout élément de 𝒢 ¯ a une longueur des commutateur inférieure ou égale à 6. De plus, nous exhibons des conditions sur 𝒢 qui garantissent que les longueurs des commutateurs des éléments de 𝒢 0 sont uniformément bornées et dans ce cas pour tout g∈𝒢 0 nous montrons que cette longueur est au plus 5.

Let 𝒢 be the group of all Interval Exchange Transformations. Results of Arnoux–Fathi, Sah and Vorobets state that 𝒢 0 the subgroup of 𝒢 generated by its commutators is simple. Arnoux proved that the group 𝒢 ¯ of all Interval Exchange Transformations with flips is simple.

We establish that the commutator length is at most 6 for any element of 𝒢 ¯. Moreover, we give conditions on 𝒢 that guarantee that the commutator lengths of the elements of 𝒢 0 are uniformly bounded, and in this case for any g∈𝒢 0 this length is at most 5.

Reçu le :
Accepté le :
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DOI : 10.5802/aif.3502
Classification : 57S30, 37E05, 20F12
Keywords: Interval exchange transformation, Commutator, Perfect groups, Commutator length
Mot clés : échanges d’intervalles, commutateur, groupes parfaits, longueur des commutateur

Guelman, Nancy 1 ; Liousse, Isabelle 2

1 IMERL, Facultad de Ingeniería Universidad de la República C.C. 30, Montevideo (Uruguay)
2 Univ. Lille, CNRS UMR 8524 - Laboratoire Paul Painlevé F-59000 Lille (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Guelman, Nancy; Liousse, Isabelle. Uniform perfectness for Interval Exchange Transformations with or without Flips. Annales de l'Institut Fourier, Tome 72 (2022) no. 4, pp. 1477-1501. doi : 10.5802/aif.3502. https://aif.centre-mersenne.org/articles/10.5802/aif.3502/

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