Groups of interval exchange transformations
François Dahmani1
1 Université Grenoble Alpes, Institut Fourier, F-38000 Grenoble, France
Winter Braids Lecture Notes, Volume 6 (2019), Talk no. 1, 22 p.

This is a survey on subgroups of the group of interval exchage transformations. We review definitions and a few properties of the groups of interval exchange transformations. We give examples of subgroups, and obstructions to realise certain subgroups.

Published online:
DOI: 10.5802/wbln.27
François Dahmani 1

1 Université Grenoble Alpes, Institut Fourier, F-38000 Grenoble, France 
@article{WBLN_2019__6__A1_0,
     author = {Fran\c{c}ois Dahmani},
     title = {Groups of interval exchange transformations},
     journal = {Winter Braids Lecture Notes},
     note = {talk:1},
     publisher = {Winter Braids School},
     volume = {6},
     year = {2019},
     doi = {10.5802/wbln.27},
     mrnumber = {3974474},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/wbln.27/}
}
TY  - JOUR
AU  - François Dahmani
TI  - Groups of interval exchange transformations
JO  - Winter Braids Lecture Notes
N1  - talk:1
PY  - 2019
VL  - 6
PB  - Winter Braids School
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/wbln.27/
DO  - 10.5802/wbln.27
LA  - en
ID  - WBLN_2019__6__A1_0
ER  - 
%0 Journal Article
%A François Dahmani
%T Groups of interval exchange transformations
%J Winter Braids Lecture Notes
%Z talk:1
%D 2019
%V 6
%I Winter Braids School
%U https://proceedings.centre-mersenne.org/articles/10.5802/wbln.27/
%R 10.5802/wbln.27
%G en
%F WBLN_2019__6__A1_0
François Dahmani. Groups of interval exchange transformations. Winter Braids Lecture Notes, Volume 6 (2019), Talk no. 1, 22 p. doi : 10.5802/wbln.27. https://proceedings.centre-mersenne.org/articles/10.5802/wbln.27/

[1] Pierre Arnoux, Echanges d’intervalles et flots sur les surfaces. L’Enseignement Mathématique, 29, 5-38 (1981). | DOI | Zbl

[2] Artur Avila, Giovanni Forni, Weak mixing for interval exchange transformations and translation flows, Ann. Math. (2) 165, No. 2, 637-664 (2007). | DOI | MR | Zbl

[3] Michael Boshernitzan, A condition for minimal interval exchange transformation to be uniquely ergodic, Duke J. of Math. 52 (3) (1985), 723-752. | DOI | MR | Zbl

[4] Michael Boshernitzan, Subgroup of interval exchanges generated by torsion elements and rotations. Proc. Am. Math. Soc. 144, No. 6, 2565-2573 (2016). | DOI | MR | Zbl

[5] Tullio Ceccherini-Silberstein, Michel Coornaert, Cellular automata and groups, Springer Monographs in Mathematics (2010) | DOI | Zbl

[6] Yves de Cornulier, Realizations of groups of piecewise continuous transformations of the circle, arXiv:1902.0699 | DOI | MR | Zbl

[7] Yves de Cornulier, Groupes pleins-topologiques, d’après Matui, Juschenko-Monod, Séminaire Bourbaki, volume 2012/2013, exposés 1059-1073 Astérisque 361 (2014) | DOI

[8] Yves de Cornulier, Commensurating actions for groups of piecewise continuous transformations, arXiv:1803.08572. | DOI | MR | Zbl

[9] François Dahmani, Koji Fujiwara, Vincent Guirardel, Free groups of interval exchange transformations are rare. Groups Geom. Dyn. 7 (2013) no. 4, 883–910. | DOI | MR | Zbl

[10] François Dahmani, Koji Fujiwara, Vincent Guirardel, Solvable groups of interval exchange transformations, Ann. Fac. Sci. Toulouse. | DOI | MR | Zbl

[11] Damien Gaboriau, Gilbert Levitt, and Frédéric Paulin, Pseudogroups of isometries of R and Rips’ theorem on free actions on ℝ-trees. Israel J. Math. 87 (1994) no. 1-3, 403–428. | DOI | MR | Zbl

[12] Arek Goetz, Piecewise isometries – an emerging area in dynamical systems. Grabner, Peter (ed.) et al., Fractals in Graz 2001. Analysis, dynamics, geometry, stochastics. Proceedings of the conference, Graz, Austria, June 2001. Basel: Birkhäuser. Trends in Mathematics. 135-144 (2003). | DOI | Zbl

[13] Rostislav Grigorchuk, Konstantin Medynets, On Algebraic Properties of Topological Full Groups, Sb. Math. 205, No. 6, 843-861 (2014); | DOI | MR | Zbl

[14] Boris Hasselblatt, Anatole Katok, Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995. xviii+802 pp. | DOI | Zbl

[15] Pierre de la Harpe, Mesures finiment additives et paradoxes. In: Autour du Centenaire Lebesgue. Panor. Synthèses, vol. 18, pp. 39–61. Soc. Math. France, Paris (2004) | DOI | Zbl

[16] Hideki Imanishi, On codimension one foliations defined by closed one-forms with singularities. J. Math. Kyoto Univ. 19 (1979), no. 2, 285–291. | DOI | MR | Zbl

[17] Kate Juschenko, Nicolas Monod, Cantor systems, piecewise translations and simple amenable groups, Annals of Math, 178, no. 2 (2013), 775-787. | DOI | MR | Zbl

[18] Kate Juschenko, Nicolas Matte-Bon, Nicolas Monod, Mikael de la Salle, Extensive amenability and an application to interval exchanges, Ergodic Theory Dynam. Systems. 38, no.1 195–219 (2018). | DOI | MR

[19] Kate Juschenko Amenability of discrete groups by examples. (2015). | DOI

[20] Anatole B. Katok, Anatoly M. Stepin, Approximations in ergodic theory, Usp. Mat. Nauk22(5), 81–106 (1967) | DOI | MR | Zbl

[21] Yitzhak. Katznelson, Benjamin Weiss, A simple proof of some ergodic theorems. Israel J. Math. 42 (1982), no. 4, 291–296. | DOI | MR | Zbl

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

[23] Marc Krasner and Léo Kaloujnine, Produit complet des groupes de permutations et le problème d’extension de groupes III, Acta Sci. Math. Szeged 14, pp. 69-82 (1951) | DOI | Zbl

[24] Octave Lacourte, Abelianization of some groups of interval exchanges, arXiv:2009.07595 | DOI | MR

[25] Isabelle Liousse, Nancy Guelman, Distortion in groups of Affine Interval Exchange transformations, Groups Geom. Dyn. 13, No. 3, 795-819 (2019). | DOI | MR | Zbl

[26] W. Magnus, A. Karrass, D. Solitar Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, Dover Publications; 2nd Revised ed. edition (2004). | DOI | MR | Zbl

[27] Howard Masur, Interval exchange transformations and measured foliations, Ann. Math. (2) 115, 169-200 (1982). | DOI | MR | Zbl

[28] John Meier Groups, Graphs, and Trees, vol 73 of London Mathematical Society Student Texts, Cambridge University Press (2008). | DOI

[29] Christopher Novak, Discontinuity growth of interval exchange maps. J Mod. Dyn. 3(3) 379–405 (2009). | DOI | MR | Zbl

[30] Christopher Novak, Continuous interval exchange actions. Algebr. Geom. Topol. 10, No. 3, 1609-1625 (2010). | DOI | MR | Zbl

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

[32] Chih-Han Sah, Scissors congruences of the interval, Preprint (State Univ. New York, Stony Brook, NY, 1981) | DOI | Zbl

[33] Nóra Szoke, A Tits alternative for topological full groups, arXiv:1808.09882 | DOI | MR | Zbl

[34] Nóra Szoke, Extensive amenability and a Tits alternative for topological full groups, Thèse de Doctorat de l’École Polytechnique Fédérale de Lausanne, 2019. | DOI | Zbl

[35] William Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. Math. (2) 115, 201-242 (1982). | DOI | MR | Zbl

[36] William Veech, The metric theory of interval exchange transformations. III: The Sah- Arnoux-Fathi invariant. Am. J. Math. 106, 1389-1422 (1984). | DOI | MR | Zbl

[37] Yaroslav Vorobets, On the Commutator Group of the Groupof Interval Exchange Transformations, Proceedings of the Steklov Institute of Mathematics, 2017, Vol. 297, pp. 285–296. | DOI | Zbl

[38] Christophe Yoccoz, Echanges d’intervalles, cours au College de France, 2005. https://www.college-de-france.fr/media/jean-christophe-yoccoz/UPL8726_yoccoz05.pdf | DOI

Cited by Sources: