Smooth perfectness for the group of diffeomorphisms

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September 2013, 5(3): 281-294. doi: 10.3934/jgm.2013.5.281

Smooth perfectness for the group of diffeomorphisms

Stefan Haller ^1, , Tomasz Rybicki ^2, and Josef Teichmann ^3,

1.	Department of Mathematics, University of Vienna, Nordbergstraße 15, A-1090, Vienna, Austria
2.	Faculty of applied mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland
3.	Department of Mathematics, ETH Zürich, Rämistrasse 10, 8092 Zürich, Switzerland

Received March 2013 Published September 2013

Abstract
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Given a result of Herman, we provide a new elementary proof of the fact that the connected component of the identity in the group of compactly supported diffeomorphisms is perfect and hence simple. Moreover, we show that every diffeomorphism $g$, which is sufficiently close to the identity, can be represented as a product of four commutators, $g=[h_1,k_1]\circ\cdots\circ[h_4,k_4]$, where the factors $h_i$ can be chosen to depend smoothly on $g$, and $k_i=\exp(X_i)$ are flows at time one of complete vector fields $X_i$ which are independent of $g$.

Keywords: convenient calculus, perfect group, simple group, fragmentation, Diffeomorphism group, foliation..

Mathematics Subject Classification: 58D0.

Citation: Stefan Haller, Tomasz Rybicki, Josef Teichmann. Smooth perfectness for the group of diffeomorphisms. Journal of Geometric Mechanics, 2013, 5 (3) : 281-294. doi: 10.3934/jgm.2013.5.281

References:

[1]	A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Commentarii Mathematici Helvetici, 53 (1978), 174-227. doi: 10.1007/BF02566074.
[2]	A. Banyaga, "The Structure of Classical Diffeomorphism Groups," Kluwer Academic Publishers Group, Dordrecht, 1997.
[3]	D. Burago, S. Ivanov and L. Polterovich, Conjugation-invariant norms on groups of geometric origin, Groups of diffeomorphisms, Adv. Stud. Pure Math, Math. Soc. Japan, Tokyo, 52 (2008), 221-250.
[4]	R. D. Edwards and R. C. Kirby, Deformations of spaces of imbeddings, Annals of Mathematics, 93 (1971), 63-88. doi: 10.2307/1970753.
[5]	D. B. A. Epstein, The simplicity of certain groups of homeomorphisms, Compositio Math., 22 (1970), 165-173.
[6]	D. B. A. Epstein, Commutators of $C^\infty$-diffeomorphisms. Appendix to: "A curious remark concerning the geometric transfer map'' by John N. Mather, Commentarii Mathematici Helvetici, 59 (1984), 111-122. doi: 10.1007/BF02566339.
[7]	S. Haller and J. Teichmann, Smooth perfectness through decomposition of diffeomorphisms into fiber preserving ones, Annals of Global Analysis and Geometry, 23 (2003), 53-63. doi: 10.1023/A:1021280213742.
[8]	M. R. Herman, Simplicité du groupe des difféomorphismes de classe $C^\infty$, isotopes à l'identité, du tore de dimension n, C. R. Acad. Sci. Paris Sr. A, 273 (1971), 232-234.
[9]	M. R. Herman, Sur le groupe des difféomorphismes du tore, Annales de L'Institut Fourier (Grenoble), 23 (1973), 75-86. doi: 10.5802/aif.457.
[10]	M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics 33, Springer, 1976.
[11]	A. Kriegl and P. W. Michor, "The Convenient Setting of Global Analysis," Mathematical Surveys and Monographs 53, American Mathematical Society, 1997.
[12]	J. N. Mather, The vanishing of the homology of certain groups of homeomorphisms, Topology, 10 (1071), 297-298.
[13]	J. N. Mather, Commutators of diffeomorphisms, Comment. Math. Helv., 49 (1974), 512-528.
[14]	J. N. Mather, Commutators of diffeomorphisms. II, Comment. Math. Helv., 50 (1975), 33-40.
[15]	J. N. Mather, A curious remark concerning the geometric transfer map, Commentarii Mathematici Helvetici, 59 (1984), 86-110. doi: 10.1007/BF02566338.
[16]	J. Milnor, "Morse Theory," Annals of Mathematics Studies 51, Princeton University Press, Princeton, N.J. 1963.
[17]	S. P. Novikov, Multivalued functions and functionals. An analogue of the Morse theory, Dokl. Akad. Nauk SSSR, 260 (1981), 31-35. English translation: Soviet Math. Dokl., 24 (1981), 222-226 (1982).
[18]	A. V. Pajitnov, "Circle-valued Morse Theory," de Gruyter Studies in Mathematics 32, Walter de Gruyter & Co., Berlin, 2006. doi: 10.1515/9783110197976.
[19]	T. Rybicki, The identity component of the leaf preserving diffeomorphism group is perfect, Monatshefte fr Mathematik, 120 (1995), 289-305. doi: 10.1007/BF01294862.
[20]	T. Rybicki, Commutators of contactomorphisms, Advances in Mathematics, 225 (2010), 3291-3326. doi: 10.1016/j.aim.2010.06.004.
[21]	T. Rybicki, Locally continuously perfect groups of homeomorphisms, Annals of Global Analysis and Geometry, 40 (2011), 191-202. doi: 10.1007/s10455-011-9253-5.
[22]	W. Thurston, Foliations and groups of diffeomorphisms, Bulletin of the American Mathematical Society, 80 (1974), 304-307. doi: 10.1090/S0002-9904-1974-13475-0.
[23]	T. Tsuboi, On $2$-cycles of B Diff$(S^1)$ which are represented by foliated $S^1$-bundles over $T^2$, Ann. Inst. Fourier, 31 (1981), 1-59.
[24]	T. Tsuboi, On the uniform perfectness of diffeomorphism groups, Groups of diffeomorphisms, 505-524, Adv. Stud. Pure Math. 52, Math. Soc. Japan, Tokyo, 2008.
[25]	T. Tsuboi, On the uniform simplicity of diffeomorphism groups, Differential geometry, 43-55, World Sci. Publ., Hackensack, NJ, 2009. doi: 10.1142/9789814261173_0004.
[26]	T. Tsuboi, On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds, Commentarii Mathematici Helvetici, 87 (2012), 141-185. doi: 10.4171/CMH/251.
[27]	J.-C. Yoccoz, Petits diviseurs en dimension $1$, Astérisque, 231 (1995).

show all references

References:

[1]	A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Commentarii Mathematici Helvetici, 53 (1978), 174-227. doi: 10.1007/BF02566074.
[2]	A. Banyaga, "The Structure of Classical Diffeomorphism Groups," Kluwer Academic Publishers Group, Dordrecht, 1997.
[3]	D. Burago, S. Ivanov and L. Polterovich, Conjugation-invariant norms on groups of geometric origin, Groups of diffeomorphisms, Adv. Stud. Pure Math, Math. Soc. Japan, Tokyo, 52 (2008), 221-250.
[4]	R. D. Edwards and R. C. Kirby, Deformations of spaces of imbeddings, Annals of Mathematics, 93 (1971), 63-88. doi: 10.2307/1970753.
[5]	D. B. A. Epstein, The simplicity of certain groups of homeomorphisms, Compositio Math., 22 (1970), 165-173.
[6]	D. B. A. Epstein, Commutators of $C^\infty$-diffeomorphisms. Appendix to: "A curious remark concerning the geometric transfer map'' by John N. Mather, Commentarii Mathematici Helvetici, 59 (1984), 111-122. doi: 10.1007/BF02566339.
[7]	S. Haller and J. Teichmann, Smooth perfectness through decomposition of diffeomorphisms into fiber preserving ones, Annals of Global Analysis and Geometry, 23 (2003), 53-63. doi: 10.1023/A:1021280213742.
[8]	M. R. Herman, Simplicité du groupe des difféomorphismes de classe $C^\infty$, isotopes à l'identité, du tore de dimension n, C. R. Acad. Sci. Paris Sr. A, 273 (1971), 232-234.
[9]	M. R. Herman, Sur le groupe des difféomorphismes du tore, Annales de L'Institut Fourier (Grenoble), 23 (1973), 75-86. doi: 10.5802/aif.457.
[10]	M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics 33, Springer, 1976.
[11]	A. Kriegl and P. W. Michor, "The Convenient Setting of Global Analysis," Mathematical Surveys and Monographs 53, American Mathematical Society, 1997.
[12]	J. N. Mather, The vanishing of the homology of certain groups of homeomorphisms, Topology, 10 (1071), 297-298.
[13]	J. N. Mather, Commutators of diffeomorphisms, Comment. Math. Helv., 49 (1974), 512-528.
[14]	J. N. Mather, Commutators of diffeomorphisms. II, Comment. Math. Helv., 50 (1975), 33-40.
[15]	J. N. Mather, A curious remark concerning the geometric transfer map, Commentarii Mathematici Helvetici, 59 (1984), 86-110. doi: 10.1007/BF02566338.
[16]	J. Milnor, "Morse Theory," Annals of Mathematics Studies 51, Princeton University Press, Princeton, N.J. 1963.
[17]	S. P. Novikov, Multivalued functions and functionals. An analogue of the Morse theory, Dokl. Akad. Nauk SSSR, 260 (1981), 31-35. English translation: Soviet Math. Dokl., 24 (1981), 222-226 (1982).
[18]	A. V. Pajitnov, "Circle-valued Morse Theory," de Gruyter Studies in Mathematics 32, Walter de Gruyter & Co., Berlin, 2006. doi: 10.1515/9783110197976.
[19]	T. Rybicki, The identity component of the leaf preserving diffeomorphism group is perfect, Monatshefte fr Mathematik, 120 (1995), 289-305. doi: 10.1007/BF01294862.
[20]	T. Rybicki, Commutators of contactomorphisms, Advances in Mathematics, 225 (2010), 3291-3326. doi: 10.1016/j.aim.2010.06.004.
[21]	T. Rybicki, Locally continuously perfect groups of homeomorphisms, Annals of Global Analysis and Geometry, 40 (2011), 191-202. doi: 10.1007/s10455-011-9253-5.
[22]	W. Thurston, Foliations and groups of diffeomorphisms, Bulletin of the American Mathematical Society, 80 (1974), 304-307. doi: 10.1090/S0002-9904-1974-13475-0.
[23]	T. Tsuboi, On $2$-cycles of B Diff$(S^1)$ which are represented by foliated $S^1$-bundles over $T^2$, Ann. Inst. Fourier, 31 (1981), 1-59.
[24]	T. Tsuboi, On the uniform perfectness of diffeomorphism groups, Groups of diffeomorphisms, 505-524, Adv. Stud. Pure Math. 52, Math. Soc. Japan, Tokyo, 2008.
[25]	T. Tsuboi, On the uniform simplicity of diffeomorphism groups, Differential geometry, 43-55, World Sci. Publ., Hackensack, NJ, 2009. doi: 10.1142/9789814261173_0004.
[26]	T. Tsuboi, On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds, Commentarii Mathematici Helvetici, 87 (2012), 141-185. doi: 10.4171/CMH/251.
[27]	J.-C. Yoccoz, Petits diviseurs en dimension $1$, Astérisque, 231 (1995).

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