Non-smoothability for a class of groups of piecewise linear homeomorphisms of the interval
Annales de l'Institut Fourier, Volume 74 (2024) no. 3, pp. 1095-1108.

For a certain class of groups of piecewise linear homeomorphisms of the interval, we prove that they admit no sufficiently regular faithful action on the line. Building on previous work of Brum, Matte Bon, Rivas, and the author, the new ingredient is an observation from a recent work of Hyde and Tatch Moore, which allows to reduce the problem to the case of the circle, and then apply Herman–Yoccoz theory.

Pour une certaine classe de groupes d’homéomorphismes linéaires par morceaux de l’intervalle, on montre qu’ils n’admettent aucune action fidèle sur la droite suffisamment régulière. En s’appuyant sur un travail précédent de Brum, Matte Bon, Rivas, et l’auteur, le nouvel ingrédient est une observation introduite dans un travail récent de Hyde et Tatch Moore, qui permet de ramener le problème aux actions sur le cercle, et ensuite appliquer la théorie d’Herman–Yoccoz.

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DOI: 10.5802/aif.3626
Classification: 37C85, 57M60, 37E05, 37E10
Keywords: Group actions on the real line, locally moving groups, groups of piecewise linear homeomorphisms, Herman–Yoccoz theory, smoothability
Mot clés : Groupes agissant sur la droite, groupes localement mobiles, groupes d’homéomorphismes linéaires par morceaux, théorie d’Herman–Yoccoz, lissabilité

Triestino, Michele 1

1 Institut de Mathématiques de Bourgogne (IMB, UMR CNRS 5584) Université de Bourgogne 9 av. Alain Savary, 21000 Dijon (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Triestino, Michele. Non-smoothability for a class of groups of piecewise linear homeomorphisms of the interval. Annales de l'Institut Fourier, Volume 74 (2024) no. 3, pp. 1095-1108. doi : 10.5802/aif.3626. https://aif.centre-mersenne.org/articles/10.5802/aif.3626/

[1] Adouani, Abdelhamid; Marzougui, Habib Sur les homéomorphismes du cercle de classe PC r par morceaux (r1) qui sont conjugués C r par morceaux aux rotations irrationnelles, Ann. Inst. Fourier, Volume 58 (2008) no. 3, pp. 755-775 | DOI | MR | Zbl

[2] Bieri, Robert; Strebel, Ralph On groups of PL-homeomorphisms of the real line, Mathematical Surveys and Monographs, 215, American Mathematical Society, 2016, xvii+174 pages | DOI | MR | Zbl

[3] Bleak, Collin; Bowman, Hannah; Gordon Lynch, Alison; Graham, Garrett; Hughes, Jacob; Matucci, Francesco; Sapir, Eugenia Centralizers in the R. Thompson group V n , Groups Geom. Dyn., Volume 7 (2013) no. 4, pp. 821-865 | DOI | MR | Zbl

[4] Bonatti, Christian; Lodha, Yash; Triestino, Michele Hyperbolicity as an obstruction to smoothability for one-dimensional actions, Geom. Topol., Volume 23 (2019) no. 4, pp. 1841-1876 | DOI | MR | Zbl

[5] Brin, Matthew G.; Guzmán, Fernando Automorphisms of generalized Thompson groups, J. Algebra, Volume 203 (1998) no. 1, pp. 285-348 | DOI | MR | Zbl

[6] Brum, Joaquín; Bon, Nicolás Matte; Rivas, Cristóbal; Triestino, Michele Locally moving groups acting on the line and -focal actions (2021) (https://arxiv.org/abs/2104.14678)

[7] Dzhalilov, Akhtam; Liousse, Isabelle Circle homeomorphisms with two break points, Nonlinearity, Volume 19 (2006) no. 8, pp. 1951-1968 | DOI | MR | Zbl

[8] Ghazouani, Selim; Ulcigrai, Corinna A priori bounds for GIETs, affine shadows and rigidity of foliations in genus 2 (2021) (https://arxiv.org/abs/2106.03529)

[9] Ghys, Étienne Groupes d’homéomorphismes du cercle et cohomologie bornée, The Lefschetz centennial conference, Part III (Mexico City, 1984) (Contemporary Mathematics), Volume 58, American Mathematical Society, 1987, pp. 81-106 | DOI | MR | Zbl

[10] Ghys, Étienne; Sergiescu, Vlad Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helv., Volume 62 (1987) no. 2, pp. 185-239 | DOI | MR | Zbl

[11] Herman, Michael-Robert Sur la conjugaison différentiable des difféomorphismes ducercle à des rotations, Publ. Math., Inst. Hautes Étud. Sci. (1979) no. 49, pp. 5-233 | DOI | MR | Zbl

[12] Hmili, Hadda; Liousse, Isabelle Dynamique des échanges d’intervalles des groupes de Higman-Thompson V r,m , Ann. Inst. Fourier, Volume 64 (2014) no. 4, pp. 1477-1491 | DOI | Zbl

[13] Hyde, James; Moore, Justin Tatch Subgroups of PL + I which do not embed into Thompson’s group F, Groups Geom. Dyn., Volume 17 (2023) no. 2, pp. 533-554 | DOI | MR | Zbl

[14] Khanin, Khanin; Teplinsky, Alexey Herman’s theory revisited, Invent. Math., Volume 178 (2009) no. 2, pp. 333-344 | DOI | MR | Zbl

[15] Khanin, Konstantin; Kocić, Saša Rigidity for a class of generalized interval exchange transformations, Dynamical systems, ergodic theory, and probability. In memory of Kolya Chernov. Conference dedicated to the memory of Nikolai Chernov, University of Alabama at Birmingham, Birmingham, AL, USA, May 18–20, 2015, American Mathematical Society, 2017, pp. 161-167 | DOI | Zbl

[16] Kim, Sang-hyun; Koberda, Thomas; Lodha, Yash Chain groups of homeomorphisms of the interval, Ann. Sci. Éc. Norm. Supér., Volume 52 (2019) no. 4, pp. 797-820 | DOI | MR | Zbl

[17] Liousse, Isabelle Nombre de rotation, mesures invariantes et ratio set des homéomorphismes affines par morceaux du cercle, Ann. Inst. Fourier, Volume 55 (2005) no. 2, pp. 431-482 | DOI | Numdam | MR | Zbl

[18] Lodha, Yash Coherent actions by homeomorphisms on the real line or an interval, Isr. J. Math., Volume 235 (2020) no. 1, pp. 183-212 | DOI | MR | Zbl

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