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Abstrakty
It is shown that in some generic cases the identity component of the group of leaf preserving diffeomorphisms (with not necessarily compact support) on a foliated open manifold is perfect. Next, it is proved that it is also bounded, i.e. bounded with respect to any bi-invariant metric. It follows that the group is uniformly perfect as well.
Czasopismo
Rocznik
Tom
Strony
511--520
Opis fizyczny
Bibliogr. 19 poz.
Twórcy
autor
autor
autor
- AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, Poland, kowalik@wms.mat.agh.edu.pl
Bibliografia
- [1] K. Abe, K. Fukui, Commutators of C1-diffeomorphisms preserving a submanifold, J. Math. Soc. Japan 61 (2009), 427–436.
- [2] R.D. Anderson, On homeomorphisms as products of a given homeomorphism and its inverse, Topology of 3-manifolds, M. Fort (ed.), Prentice-Hall, 1961, 231–237.
- [3] A. Banyaga, The structure of classical diffeomorphism groups, Mathematics and Its Applications, 400, Kluwer Academic Publishers Group, Dordrecht, 1997.
- [4] D. Burago, S. Ivanov, L. Polterovich, Conjugation invariant norms on groups of geometric origin, Adv. Stud. Pure Math. 52, Groups of Diffeomorphisms (2008), 221–250.
- [5] M.R. Herman, Sur le groupe des difféomorphismes du tore, Ann. Inst. Fourier (Grenoble) 23 (1973) 2, 75–86.
- [6] M.W. Hirsch, Differential topology, Springer-Verlag, New York, 1976.
- [7] J. Lech, T. Rybicki, Groups of Cr;s-diffeomorphisms related to a foliation, [in:] Geometry and Topology of Manifolds, Banach Center Publ. 76 (2007), 437–450.
- [8] J.N. Mather, The vanishing of the homology of certain groups of homeomorphisms, Topology 10 (1971), 297–298.
- [9] J.N. Mather, Commutators of diffeomorphisms, Comment. Math. Helv. 49 (1974), 512–528.
- [10] J.N. Mather, Commutators of diffeomorphisms, II, Comment. Math. Helv. 50 (1975), 33–40.
- [11] J.N. Mather, Commutators of diffeomorphisms, III: a group which is not perfect, Comment. Math. Helv. 60 (1985), 122–124.
- [12] D. McDuff, The lattice of normal subgroups of the group of diffeomorphisms or homeomorphisms of an open manifold, J. London Math. Soc. 18 (1978), 353–364.
- [13] T. Rybicki, The identity component of the leaf preserving diffeomorphism group is perfect, Monatsh. Math. 120 (1995), 289–305.
- [14] T. Rybicki, On foliated, Poisson and Hamiltonian diffeomorphisms, Differential Geom. Appl. 15 (2001), 33–46.
- [15] T. Rybicki, Boundedness of certain automorphism groups of an open manifold, Geom. Dedicata 151 (2011) 1, 175–186.
- [16] P.A. Schweitzer, Normal subgroups of diffeomorphism and homeomorphism groups of Rn and other open manifolds, Ergodic Theory and Dynamical Systems, available on CJO, doi:10.1017/S0143385710000659.
- [17] W. Thurston, Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974), 304–307.
- [18] T. Tsuboi, On the group of foliation preserving diffeomorphisms, Foliations 2005, World Scientific, Singapore (2006), 411–430.
- [19] T. Tsuboi, On the uniform perfectness of diffeomorphism groups, Adv. Stud. Pure Math. 52, Groups of Diffeomorphisms (2008), 505–524.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-AGHS-0004-0008