Abstract
We show that the identity component of the group of homeomorphisms that preserve all leaves of a \({\mathbb R}^{d}\)-tilable lamination is simple. Moreover, in the one dimensional case, we show that this group is uniformly perfect. We obtain similar results for homeomorphisms preserving the vertical structure.
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References
Aliste-Prieto, J.: Translation numbers for a class of maps arising from one-dimensional quasicrystals. Erg. Theo. Dyn. Sys. 30 (Issue 2), 565–594 (2010)
Anderson, R.D.: On homeomorphisms as products of conjugates of a given homeomorphism and its inverse. Topology of 3-manifolds and related topics, Proc. The Univ. of Georgia Institute, pp. 231–234. Prentice-Hall, Englewood Cliffs (1961)
Bellissard, J., Benedetti, R., Gambaudo, J.-M.: Spaces of tilings, finite telescopic approximations and gap-labeling. Commun. Math. Phys. 261(1), 1–41 (2006)
Ben Ami, E., Rubin, M.: On the reconstruction problem for factorizable homeomorphism groups and foliated manifolds. Topol. Appl. 157(9), 1664–1679 (2010)
Bounemoura, A.: Simplicité des groupes de transformations de surfaces. Ensaios Matemáticos 14. Sociedade Brasileira de Matemática, Rio de Janeiro pp. ii+147 (2008)
Edwards, R., Kirby, R.C.: Deformations of spaces of imbeddings. Ann. Math. (2) 93, 63–88 (1971)
Epstein, D.B.A.: The simplicity of certain group of homeomorphisms. Comp. Math. 22, 165–173 (1970)
Fisher, G.M.: On the group of all homeomorphisms of a manifold. Trans. Am. Math. Soc. 97, 193–212 (1960)
Fukui, K.: Commutator length of leaf preserving diffeomorphisms. Pub. Res. Inst. Math. Sci. 48(3), 615–622 (2012)
Herman, M.R.: Sur le groupe des difféomorphismes du tore. Ann. Inst. Fourier (Grenoble) 23, 75–86 (1973)
Kellendonk, J.: Pattern-equivariant functions and cohomology. J. Phys. A 36(21), 5765–5772 (2003)
Kwapisz, J.: Topological friction in aperiodic minimal \(\mathbb{R}^m\)-actions. Fund. Math. 207(2), 175–178 (2010)
Mather, J.N.: Commutators of diffeomorphisms. Comment. Math. Helv. I 49,512–528 (1974); II 50, 33–40 (1975); III 60, 122–124 (1985)
Priebe Frank, N., Sadun, L.: Fusion tilings with infinite local complexity. Topol. Proc. 43, 235–276 (2014)
Rybicki, T.: The identity component of the leaf preserving diffeomorphism group is perfect. Monast. Math. 120, 289–305 (1995)
Tsuboi, T.,On the group of foliation preserving diffeomorphisms. In: Walczak, P. et al. (eds.) Foliationsed 2005. pp. 411–430. World Scientific, Singapore (2006)
Tsuboi, T.: On the uniform perfectness of diffeomorphism groups. In: Penner, R., Kotschick, D., Tsuboi, T., Kawazumi, N., Kitan, T., Mitsumatsu, Y. (eds.) Groups of Diffeomorphisms. Advanced Studies in Pure Mathematics, vol. 52, pp. 505–524. Mathematical Society of Japan, Tokyo (2008)
Thurston, W.: Foliations and groups of diffeomorphisms. Bull. Am. Math. Soc. 80, 304–307 (1974)
Acknowledgments
It is a pleasure for S. Petite to acknowledge A. Rivière for all the discussions on the subtleties of the Schoenflies Theorem. J. Aliste-Prieto acknowledges financial support from Fondecyt Iniciación 11121510 and Anillo DySyRf ACT-1103. S. Petite acknowledges financial support from the ANR SUBTILE 0879. This work is part of the program MathAmSud DYSTIL 12Math-02
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Communicated by A. Constantin.
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Aliste-Prieto, J., Petite, S. On the simplicity of homeomorphism groups of a tilable lamination. Monatsh Math 181, 285–300 (2016). https://doi.org/10.1007/s00605-016-0921-1
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DOI: https://doi.org/10.1007/s00605-016-0921-1