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.
@article{WBLN_2019__6__A1_0, author = {Dahmani, Fran\c{c}ois}, title = {Groups of interval exchange transformations}, booktitle = {Winter Braids IX (Reims, 2019)}, series = {Winter Braids Lecture Notes}, note = {talk:1}, pages = {1--22}, publisher = {Winter Braids School}, year = {2019}, doi = {10.5802/wbln.27}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/wbln.27/} }
TY - JOUR AU - Dahmani, François TI - Groups of interval exchange transformations BT - Winter Braids IX (Reims, 2019) AU - Collectif T3 - Winter Braids Lecture Notes N1 - talk:1 PY - 2019 SP - 1 EP - 22 PB - Winter Braids School UR - http://archive.numdam.org/articles/10.5802/wbln.27/ DO - 10.5802/wbln.27 LA - en ID - WBLN_2019__6__A1_0 ER -
%0 Journal Article %A Dahmani, François %T Groups of interval exchange transformations %B Winter Braids IX (Reims, 2019) %A Collectif %S Winter Braids Lecture Notes %Z talk:1 %D 2019 %P 1-22 %I Winter Braids School %U http://archive.numdam.org/articles/10.5802/wbln.27/ %R 10.5802/wbln.27 %G en %F WBLN_2019__6__A1_0
Dahmani, François. Groups of interval exchange transformations, dans Winter Braids IX (Reims, 2019), Winter Braids Lecture Notes (2019), Exposé no. 1, 22 p. doi : 10.5802/wbln.27. http://archive.numdam.org/articles/10.5802/wbln.27/
[1] Pierre Arnoux, Echanges d’intervalles et flots sur les surfaces. L’Enseignement Mathématique, 29, 5-38 (1981). | 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
[7] Yves de Cornulier, Groupes pleins-topologiques, d’après Matui, Juschenko-Monod, Séminaire Bourbaki, volume 2012/2013, exposés 1059-1073 Astérisque 361 (2014)
[8] Yves de Cornulier, Commensurating actions for groups of piecewise continuous transformations, arXiv:1803.08572.
[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 and Rips’ theorem on free actions on -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 | 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) | 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).
[20] Anatole B. Katok, Anatoly M. Stepin, Approximations in ergodic theory, Usp. Mat. Nauk22(5), 81–106 (1967) | DOI | 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) | Zbl
[24] Octave Lacourte, Abelianization of some groups of interval exchanges, arXiv:2009.07595
[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). | 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).
[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)
[33] Nóra Szoke, A Tits alternative for topological full groups, arXiv:1808.09882
[34] Nóra Szoke, Extensive amenability and a Tits alternative for topological full groups, Thèse de Doctorat de l’École Polytechnique Fédérale de Lausanne, 2019.
[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
Cité par Sources :