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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 |
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.
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. |
[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. |
[4] |
Y. Cornulier, Groupes pleins-topologiques (d'après Matui, Juschenko, Monod, $\dots$), Astérisque No. 361 (2014), Exp. No. 1064, ⅷ, 183–223. |
[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. |
[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. |
[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. |
[11] |
M. Keane,
Interval exchange transformations, Math. Z., 141 (1975), 25-31.
doi: 10.1007/BF01236981. |
[12] |
G. Mackey,
Point realizations of transformation groups, Illinois J. Math., 6 (1962), 327-335.
doi: 10.1215/ijm/1255632330. |
[13] |
C. F. Novak,
Discontinuity-growth of interval-exchange maps, J. Mod. Dyn., 3 (2009), 379-405.
doi: 10.3934/jmd.2009.3.379. |
[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. |
[15] |
J. B. Wagoner,
Delooping classifying spaces in algebraic K-theory, Topology, 11 (1972), 349-370.
doi: 10.1016/0040-9383(72)90031-6. |
show all references
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. |
[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. |
[4] |
Y. Cornulier, Groupes pleins-topologiques (d'après Matui, Juschenko, Monod, $\dots$), Astérisque No. 361 (2014), Exp. No. 1064, ⅷ, 183–223. |
[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. |
[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. |
[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. |
[11] |
M. Keane,
Interval exchange transformations, Math. Z., 141 (1975), 25-31.
doi: 10.1007/BF01236981. |
[12] |
G. Mackey,
Point realizations of transformation groups, Illinois J. Math., 6 (1962), 327-335.
doi: 10.1215/ijm/1255632330. |
[13] |
C. F. Novak,
Discontinuity-growth of interval-exchange maps, J. Mod. Dyn., 3 (2009), 379-405.
doi: 10.3934/jmd.2009.3.379. |
[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. |
[15] |
J. B. Wagoner,
Delooping classifying spaces in algebraic K-theory, Topology, 11 (1972), 349-370.
doi: 10.1016/0040-9383(72)90031-6. |


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