Uniform perfectness for Interval Exchange Transformations with or without Flips

Guelman, Nancy; Liousse, Isabelle

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 $\overline{𝒢}$ des échanges d’intervalles avec flips est simple.

Nous établissons que tout élément de $\overline{𝒢}$ 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 $g\in {𝒢}_0$ 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 $\overline{𝒢}$ of all Interval Exchange Transformations with flips is simple.

We establish that the commutator length is at most $6$ for any element of $\overline{𝒢}$. 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 $g\in {𝒢}_0$ this length is at most $5$.

Classification : 57S30, 37E05, 20F12
Keywords: Interval exchange transformation, Commutator, Perfect groups, Commutator length
Mot clés : échanges d’intervalles, commutateur, groupes parfaits, longueur des commutateur

[1] Arnoux, Pierre 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] Arnoux, Pierre Un invariant pour les échanges d’intervalles et les flots sur les surfaces, Ph. D. Thesis, Université de Reims (1981)

[3] Banyaga, Augustin The structure of classical diffeomorphism groups, Mathematics and its Applications, 400, Kluwer Academic Publishers, 1997, xii+197 pages | DOI | MR

[4] Bounemoura, Abed Simplicité des groupes de transformations de surfaces, Ensaios Matemáticos, 14, Sociedade Brasileira de Matemática, 2008, ii+147 pages | MR

[5] Brin, Matthew G.; Squier, Craig C. Groups of piecewise linear homeomorphisms of the real line, Invent. Math., Volume 79 (1985) no. 3, pp. 485-498 | DOI | MR | Zbl

[6] Burago, Dmitri; Ivanov, Sergei 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] Dahmani, François; Fujiwara, Koji; Guirardel, Vincent Free groups of interval exchange transformations are rare, Groups Geom. Dyn., Volume 7 (2013) no. 4, pp. 883-910 | DOI | MR | Zbl

[8] Dahmani, François; Fujiwara, Koji; Guirardel, Vincent Solvable groups of interval exchange transformations (2017) (https://arxiv.org/abs/1701.00377)

[9] Dennis, Roger K.; Vaserstein, Leonid N. Commutators in linear groups, $K$-Theory, Volume 2 (1989) no. 6, pp. 761-767 | DOI | MR | Zbl

[10] Epstein, David B. A. The simplicity of certain groups of homeomorphisms, Compos. Math., Volume 22 (1970), pp. 165-173 | Numdam | MR | Zbl

[11] Farb, Benson; Franks, John 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] Gal, Światosław R.; Gismatullin, Jakub 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] Guelman, Nancy; Liousse, Isabelle Distortion in groups of affine interval exchange transformations, Groups Geom. Dyn., Volume 13 (2019) no. 3, pp. 795-819 | DOI | MR | Zbl

[14] Guelman, Nancy; Liousse, Isabelle Reversible Maps and Products of Involutions in Groups of IETS (2019) (https://arxiv.org/abs/1907.01808)

[15] Gutierrez, Carlos Smooth nonorientable nontrivial recurrence on two-manifolds, J. Differ. Equations, Volume 29 (1978) no. 3, pp. 388-395 | DOI | MR | Zbl

[16] Gutierrez, Carlos; Lloyd, Simon; Medvedev, Vladislav; Pires, Benito; Zhuzhoma, Evgeny Transitive circle exchange transformation with flips, Discrete Contin. Dyn. Syst., Volume 26 (2009) no. 1, pp. 251-263 | DOI | MR | Zbl

[17] Hubert, Pascal; Paris-Romaskevich, Olga Triangle tiling billiards and the exceptional family of their escaping trajectories: circumcenters and Rauzy gasket (2018) (https://arxiv.org/abs/1804.00181)

[18] Juschenko, Kate; Monod, Nicolas Cantor systems, piecewise translations and simple amenable groups, Ann. Math., Volume 178 (2013) no. 2, pp. 775-787 | DOI | MR | Zbl

[19] Keane, Michael Interval exchange transformations, Math. Z., Volume 141 (1975), pp. 25-31 | DOI | MR | Zbl

[20] Lacourte, Octave Abelianization of some groups of interval exchanges (2020) (https://arxiv.org/abs/2009.07595)

[21] Lacourte, Octave Signature for piecewise continuous groups (2020) (https://arxiv.org/abs/2002.12851)

[22] Linero Bas, Antonio; Soler López, Gabriel Minimal interval exchange transformations with flips, Ergodic Theory Dyn. Syst., Volume 38 (2018) no. 8, pp. 3101-3144 | DOI | MR | Zbl

[23] Masur, Howard Interval exchange transformations and measured foliations, Ann. Math., Volume 115 (1982) no. 1, pp. 169-200 | DOI | MR | Zbl

[24] Mayer, A. Trajectories on the closed orientable surfaces, Mat. Sb., N. Ser., Volume 12(54) (1943), pp. 71-84 | MR | Zbl

[25] Nogueira, Arnaldo 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] Nogueira, Arnaldo; Pires, Benito; Troubetzkoy, Serge Orbit structure of interval exchange transformations with flip, Nonlinearity, Volume 26 (2013) no. 2, pp. 525-537 | DOI | MR | Zbl

[27] Novak, Christopher F. Discontinuity-growth of interval-exchange maps, J. Mod. Dyn., Volume 3 (2009) no. 3, pp. 379-405 | DOI | MR | Zbl

[28] Novak, Christopher F. Interval exchanges that do not occur in free groups, Groups Geom. Dyn., Volume 6 (2012) no. 4, pp. 755-763 | DOI | MR | Zbl

[29] O’Farrell, Anthony G.; Short, Ian Reversibility in dynamics and group theory, London Mathematical Society Lecture Note Series, 416, Cambridge University Press, 2015, xii+281 pages | DOI

[30] Paris-Romaskevich, Olga Notes on Tilling billiards : Some thoughts and questions (2019) (http://pa-ro.net/doc/Tree-conjecture.pdf)

[31] Sah, Chih-Han Scissors congruences of the interval (1981) (preprint)

[32] Schreier, Józef; Ulam, Stanisław M. Eine Bemerkung über die Gruppe der topologisehen Abbildungen der Kreislinie auf sich selbst, Stud. Math., Volume 5 (1934), pp. 155-159 | DOI | Zbl

[33] Skripchenko, Alexandra; Troubetzkoy, Serge 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] Ulam, Stanisław M.; von Neuman, J. On the group of homeomorphisms of the surface of a sphere, Bull. Am. Math. Soc., Volume 53 (1947), p. 506

[35] Veech, William A. 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] Viana, Marcelo Ergodic theory of interval exchange maps, Rev. Mat. Complut., Volume 19 (2006) no. 1, pp. 7-100 | DOI | MR | Zbl

[37] Vorobets, Yaroslav On the commutator group of the group of interval exchange transformations, Tr. Mat. Inst. Steklova, Volume 297 (2017), pp. 313-325 | DOI | MR | Zbl