Let be the group of all Interval Exchange Transformations. Results of Arnoux–Fathi, Sah and Vorobets state that the subgroup of generated by its commutators is simple. Arnoux proved that the group of all Interval Exchange Transformations with flips is simple.
We establish that the commutator length is at most for any element of . Moreover, we give conditions on that guarantee that the commutator lengths of the elements of are uniformly bounded, and in this case for any this length is at most .
Soit le groupe des échanges d’intervalles. Des résultats d’Arnoux–Fathi, Sah et Vorobets indiquent que le sous-groupe de engendré par ses commutateurs est simple. Arnoux prouve que le groupe des échanges d’intervalles avec flips est simple.
Nous établissons que tout élément de a une longueur des commutateur inférieure ou égale à . De plus, nous exhibons des conditions sur qui garantissent que les longueurs des commutateurs des éléments de sont uniformément bornées et dans ce cas pour tout nous montrons que cette longueur est au plus .
Accepted:
Published online:
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
@article{AIF_2022__72_4_1477_0, author = {Guelman, Nancy and Liousse, Isabelle}, title = {Uniform perfectness for {Interval} {Exchange} {Transformations} with or without {Flips}}, journal = {Annales de l'Institut Fourier}, pages = {1477--1501}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {72}, number = {4}, year = {2022}, doi = {10.5802/aif.3502}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3502/} }
TY - JOUR AU - Guelman, Nancy AU - Liousse, Isabelle TI - Uniform perfectness for Interval Exchange Transformations with or without Flips JO - Annales de l'Institut Fourier PY - 2022 SP - 1477 EP - 1501 VL - 72 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3502/ DO - 10.5802/aif.3502 LA - en ID - AIF_2022__72_4_1477_0 ER -
%0 Journal Article %A Guelman, Nancy %A Liousse, Isabelle %T Uniform perfectness for Interval Exchange Transformations with or without Flips %J Annales de l'Institut Fourier %D 2022 %P 1477-1501 %V 72 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3502/ %R 10.5802/aif.3502 %G en %F AIF_2022__72_4_1477_0
Guelman, Nancy; Liousse, Isabelle. Uniform perfectness for Interval Exchange Transformations with or without Flips. Annales de l'Institut Fourier, Volume 72 (2022) no. 4, pp. 1477-1501. doi : 10.5802/aif.3502. https://aif.centre-mersenne.org/articles/10.5802/aif.3502/
[1] Échanges d’intervalles et flots sur les surfaces, Théorie ergodique (Sem., Les Plans-sur-Bex, 1980) (French) (Monographies de l’Enseignement Mathématique), Volume 29, L’Enseignement Mathématique, 1981, pp. 5-38 | MR | Zbl
[2] Un invariant pour les échanges d’intervalles et les flots sur les surfaces, Ph. D. Thesis, Université de Reims (1981)
[3] The structure of classical diffeomorphism groups, Mathematics and its Applications, 400, Kluwer Academic Publishers, 1997, xii+197 pages | DOI | MR
[4] Simplicité des groupes de transformations de surfaces, Ensaios Matemáticos, 14, Sociedade Brasileira de Matemática, 2008, ii+147 pages | MR
[5] Groups of piecewise linear homeomorphisms of the real line, Invent. Math., Volume 79 (1985) no. 3, pp. 485-498 | DOI | MR | Zbl
[6] A remark on the group of PL-homeomorphisms in dimension one, Geometric and probabilistic structures in dynamics (Contemporary Mathematics), Volume 469, American Mathematical Society, 2008, pp. 141-148 | DOI | MR | Zbl
[7] Free groups of interval exchange transformations are rare, Groups Geom. Dyn., Volume 7 (2013) no. 4, pp. 883-910 | DOI | MR | Zbl
[8] Solvable groups of interval exchange transformations (2017) (https://arxiv.org/abs/1701.00377)
[9] Commutators in linear groups, -Theory, Volume 2 (1989) no. 6, pp. 761-767 | DOI | MR | Zbl
[10] The simplicity of certain groups of homeomorphisms, Compos. Math., Volume 22 (1970), pp. 165-173 | Numdam | MR | Zbl
[11] Groups of homeomorphisms of one-manifolds. III. Nilpotent subgroups, Ergodic Theory Dyn. Syst., Volume 23 (2003) no. 5, pp. 1467-1484 | DOI | MR | Zbl
[12] Uniform simplicity of groups with proximal action, Trans. Amer. Math. Soc., Ser. B, Volume 4 (2017), pp. 110-130 (With an appendix by N. Lazarovich) | DOI | MR | Zbl
[13] Distortion in groups of affine interval exchange transformations, Groups Geom. Dyn., Volume 13 (2019) no. 3, pp. 795-819 | DOI | MR | Zbl
[14] Reversible Maps and Products of Involutions in Groups of IETS (2019) (https://arxiv.org/abs/1907.01808)
[15] Smooth nonorientable nontrivial recurrence on two-manifolds, J. Differ. Equations, Volume 29 (1978) no. 3, pp. 388-395 | DOI | MR | Zbl
[16] Transitive circle exchange transformation with flips, Discrete Contin. Dyn. Syst., Volume 26 (2009) no. 1, pp. 251-263 | DOI | MR | Zbl
[17] Triangle tiling billiards and the exceptional family of their escaping trajectories: circumcenters and Rauzy gasket (2018) (https://arxiv.org/abs/1804.00181)
[18] Cantor systems, piecewise translations and simple amenable groups, Ann. Math., Volume 178 (2013) no. 2, pp. 775-787 | DOI | MR | Zbl
[19] Interval exchange transformations, Math. Z., Volume 141 (1975), pp. 25-31 | DOI | MR | Zbl
[20] Abelianization of some groups of interval exchanges (2020) (https://arxiv.org/abs/2009.07595)
[21] Signature for piecewise continuous groups (2020) (https://arxiv.org/abs/2002.12851)
[22] Minimal interval exchange transformations with flips, Ergodic Theory Dyn. Syst., Volume 38 (2018) no. 8, pp. 3101-3144 | DOI | MR | Zbl
[23] Interval exchange transformations and measured foliations, Ann. Math., Volume 115 (1982) no. 1, pp. 169-200 | DOI | MR | Zbl
[24] Trajectories on the closed orientable surfaces, Mat. Sb., N. Ser., Volume 12(54) (1943), pp. 71-84 | MR | Zbl
[25] Almost all interval exchange transformations with flips are nonergodic, Ergodic Theory Dyn. Syst., Volume 9 (1989) no. 3, pp. 515-525 | DOI | MR | Zbl
[26] Orbit structure of interval exchange transformations with flip, Nonlinearity, Volume 26 (2013) no. 2, pp. 525-537 | DOI | MR | Zbl
[27] Discontinuity-growth of interval-exchange maps, J. Mod. Dyn., Volume 3 (2009) no. 3, pp. 379-405 | DOI | MR | Zbl
[28] Interval exchanges that do not occur in free groups, Groups Geom. Dyn., Volume 6 (2012) no. 4, pp. 755-763 | DOI | MR | Zbl
[29] Reversibility in dynamics and group theory, London Mathematical Society Lecture Note Series, 416, Cambridge University Press, 2015, xii+281 pages | DOI
[30] Notes on Tilling billiards : Some thoughts and questions (2019) (http://pa-ro.net/doc/Tree-conjecture.pdf)
[31] Scissors congruences of the interval (1981) (preprint)
[32] Eine Bemerkung über die Gruppe der topologisehen Abbildungen der Kreislinie auf sich selbst, Stud. Math., Volume 5 (1934), pp. 155-159 | DOI | Zbl
[33] On the Hausdorff dimension of minimal interval exchange transformations with flips, J. Lond. Math. Soc., Volume 97 (2018) no. 2, pp. 149-169 | DOI | MR | Zbl
[34] On the group of homeomorphisms of the surface of a sphere, Bull. Am. Math. Soc., Volume 53 (1947), p. 506
[35] Gauss measures for transformations on the space of interval exchange maps, Ann. Math., Volume 115 (1982) no. 1, pp. 201-242 | DOI | MR | Zbl
[36] Ergodic theory of interval exchange maps, Rev. Mat. Complut., Volume 19 (2006) no. 1, pp. 7-100 | DOI | MR | Zbl
[37] On the commutator group of the group of interval exchange transformations, Tr. Mat. Inst. Steklova, Volume 297 (2017), pp. 313-325 | DOI | MR | Zbl
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