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2020, 16: 59-80. doi: 10.3934/jmd.2020003

## Realizations of groups of piecewise continuous transformations of the circle

 CNRS and Univ Lyon, University Claude Bernard Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, 69622 Villeurbanne, France

Received  April 02, 2019 Revised  September 16, 2019 Published  February 2020

We study the near action of the group $\mathrm{PC}$ of piecewise continuous self-transformations of the circle. Elements of this group are only defined modulo indeterminacy on a finite subset, which raises the question of realizability: a subgroup of $\mathrm{PC}$ is said to be realizable if it can be lifted to a group of permutations of the circle.

We prove that every finitely generated abelian subgroup of $\mathrm{PC}$ is realizable. We show that this is not true for arbitrary subgroups, by exhibiting a non-realizable finitely generated subgroup of the group of interval exchanges with flips.

The group of (oriented) interval exchanges is obviously realizable (choosing the unique left-continuous representative). We show that it has only two realizations (up to conjugation by a finitely supported permutation): the left and right-continuous ones.

Citation: Yves Cornulier. Realizations of groups of piecewise continuous transformations of the circle. Journal of Modern Dynamics, 2020, 16: 59-80. doi: 10.3934/jmd.2020003
##### References:
 [1] P. Arnoux, Échanges d'intervalles et flots sur les surfaces, in Ergodic Theory (Sem., Les Plans-sur-Bex, 1980) (French), Monograph. Enseign. Math., 29, Univ. Genève, Geneva, 1981, 5–38.  Google Scholar [2] P. Arnoux, Un invariant pour les échanges d'intervalles et les flots sur les surfaces, Thèse 3e cycle, Fac. Sci. Reims, 1981. Google Scholar [3] M. Boshernitzan, Subgroup of interval exchanges generated by torsion elements and rotations, Proc. Amer. Math. Soc., 144 (2016), 2565-2573.  doi: 10.1090/proc/12958.  Google Scholar [4] Y. Cornulier, Groupes pleins-topologiques (d'après Matui, Juschenko, Monod, $\dots$), Astérisque No. 361 (2014), Exp. No. 1064, ⅷ, 183–223.  Google Scholar [5] Y. Cornulier, Commensurating actions for groups of piecewise continuous transformations, arXiv: 1803.08572, (2018). Google Scholar [6] Y. Cornulier, Near actions, arXiv: 1901.05065, (2019). Google Scholar [7] F. Dahmani, K. Fujiwara and V. Guirardel, Free groups of interval exchange transformations are rare, Groups Geom. Dyn., 7 (2013), 883-910.  doi: 10.4171/GGD/209.  Google Scholar [8] F. Dahmani, K. Fujiwara and V. Guirardel, Solvable groups of interval exchange transformations, to appear in Ann. Fac. Sci. Toulouse, arXiv: 1701.00377, 2018. Google Scholar [9] K. Juschenko and N. Monod, Cantor systems, piecewise translations and simple amenable groups, Ann. of Math. (2), 178 (2013), 775–787. doi: 10.4007/annals.2013.178.2.7.  Google Scholar [10] K. Juschenko, N. Matte Bon, N. Monod and M. de la Salle, Extensive amenability and an application to interval exchanges, Ergodic Theory Dynam. Systems, 38 (2018), 195-219.  doi: 10.1017/etds.2016.32.  Google Scholar [11] M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31.  doi: 10.1007/BF01236981.  Google Scholar [12] G. Mackey, Point realizations of transformation groups, Illinois J. Math., 6 (1962), 327-335.  doi: 10.1215/ijm/1255632330.  Google Scholar [13] C. F. Novak, Discontinuity-growth of interval-exchange maps, J. Mod. Dyn., 3 (2009), 379-405.  doi: 10.3934/jmd.2009.3.379.  Google Scholar [14] W. Scott and L. Sonneborn, Translations of infinite subsets of a group, Colloq. Math., 10 (1963), 217-220.  doi: 10.4064/cm-10-2-217-220.  Google Scholar [15] J. B. Wagoner, Delooping classifying spaces in algebraic K-theory, Topology, 11 (1972), 349-370.  doi: 10.1016/0040-9383(72)90031-6.  Google Scholar

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##### References:
 [1] P. Arnoux, Échanges d'intervalles et flots sur les surfaces, in Ergodic Theory (Sem., Les Plans-sur-Bex, 1980) (French), Monograph. Enseign. Math., 29, Univ. Genève, Geneva, 1981, 5–38.  Google Scholar [2] P. Arnoux, Un invariant pour les échanges d'intervalles et les flots sur les surfaces, Thèse 3e cycle, Fac. Sci. Reims, 1981. Google Scholar [3] M. Boshernitzan, Subgroup of interval exchanges generated by torsion elements and rotations, Proc. Amer. Math. Soc., 144 (2016), 2565-2573.  doi: 10.1090/proc/12958.  Google Scholar [4] Y. Cornulier, Groupes pleins-topologiques (d'après Matui, Juschenko, Monod, $\dots$), Astérisque No. 361 (2014), Exp. No. 1064, ⅷ, 183–223.  Google Scholar [5] Y. Cornulier, Commensurating actions for groups of piecewise continuous transformations, arXiv: 1803.08572, (2018). Google Scholar [6] Y. Cornulier, Near actions, arXiv: 1901.05065, (2019). Google Scholar [7] F. Dahmani, K. Fujiwara and V. Guirardel, Free groups of interval exchange transformations are rare, Groups Geom. Dyn., 7 (2013), 883-910.  doi: 10.4171/GGD/209.  Google Scholar [8] F. Dahmani, K. Fujiwara and V. Guirardel, Solvable groups of interval exchange transformations, to appear in Ann. Fac. Sci. Toulouse, arXiv: 1701.00377, 2018. Google Scholar [9] K. Juschenko and N. Monod, Cantor systems, piecewise translations and simple amenable groups, Ann. of Math. (2), 178 (2013), 775–787. doi: 10.4007/annals.2013.178.2.7.  Google Scholar [10] K. Juschenko, N. Matte Bon, N. Monod and M. de la Salle, Extensive amenability and an application to interval exchanges, Ergodic Theory Dynam. Systems, 38 (2018), 195-219.  doi: 10.1017/etds.2016.32.  Google Scholar [11] M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31.  doi: 10.1007/BF01236981.  Google Scholar [12] G. Mackey, Point realizations of transformation groups, Illinois J. Math., 6 (1962), 327-335.  doi: 10.1215/ijm/1255632330.  Google Scholar [13] C. F. Novak, Discontinuity-growth of interval-exchange maps, J. Mod. Dyn., 3 (2009), 379-405.  doi: 10.3934/jmd.2009.3.379.  Google Scholar [14] W. Scott and L. Sonneborn, Translations of infinite subsets of a group, Colloq. Math., 10 (1963), 217-220.  doi: 10.4064/cm-10-2-217-220.  Google Scholar [15] J. B. Wagoner, Delooping classifying spaces in algebraic K-theory, Topology, 11 (1972), 349-370.  doi: 10.1016/0040-9383(72)90031-6.  Google Scholar
Examples of graphs of elements of $\mathrm{PC}^ {\, \mathrm{\bowtie}}( \mathbf{S})$ (parameterizing the circle as an interval). The first belongs to $\mathrm{IET}^+$; the second belongs to $\mathrm{IET}^-$; the third belongs to $\mathrm{IET}^\bowtie \backslash\mathrm{IET}^\pm$. The fourth is a more "typical" element of $\mathrm{PC}^ {\, \mathrm{\bowtie}}( \mathbf{S})$. The value at breakpoints is not prescribed, as we consider group elements as defined up to finite indeterminacy
Graphs of a 132-flip and a triple flip: in each case there are two hyper-clean lifts, choosing either the endpoints denoted as circles or dots
Graphs of $u$, $v$, $w$ and $s$
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