×

Boundedness of certain automorphism groups of an open manifold. (English) Zbl 1213.22009

The main result of this paper is the following statement:
Theorem. Let \(M\) be a manifold of class \(C^r\), \(r=1,2,\dots,\infty\), \(r\neq \dim(M)+1\), which is the interior of a compact manifold \(\overline{M}\). Let \(\partial_i\), \(i=1,2,\dots,k\), be the family of all components of the boundary of \(\overline{M}\) and \(J\subset \{1,2,\dots,k\}\). Assume that \(M\) is portable (or, more general, satisfies a certain technical property described in the paper). Then the group \(\mathcal{D}_J^r(M)\) of all \(C^r\)-diffeomorphisms, which are equal to the identity on a neighborhood of \(\cup_{i\in J}\partial_i\) and can be joined with the identity by an isotopy, is bounded, uniformly perfect and has commutator length diameter at most four.

MSC:

22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
57R50 Differential topological aspects of diffeomorphisms
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abe K., Fukui K.: Commutators of C diffeomorphisms preserving a submanifold. J. Math. Soc. Japan 61, 427–436 (2009) · Zbl 1169.57029 · doi:10.2969/jmsj/06120427
[2] Anderson R.D.: On homeomorphisms as products of a given homeomorphism and its inverse. In: Fort, M. (eds) Topology of 3-manifolds, pp. 231–237. Prentice-Hall, Englewood Cliffs, NJ (1961)
[3] Banyaga A.: The structure of classical diffeomorphism groups, Mathematics and its Applications, vol. 400. Kluwer Academic Publishers Group, Dordrecht (1997) · Zbl 0874.58005
[4] Bavard C.: Longeur stables des commutateurs. Enseign. Math. 37, 109–150 (1991) · Zbl 0810.20026
[5] Burago, D., Ivanov, S., Polterovich, L.: Conjugation invariant norms on groups of geometric origin, Advanced Studies in Pures Math. 52, Groups of Diffeomorphisms, pp. 221–250 (2008) · Zbl 1222.20031
[6] Denjoy A.: Sur les courbes definies par les équations différentielles à la surface du tore. J. Math. Pures Appl. 11(9), 333–375 (1932) · Zbl 0006.30501
[7] Edwards R.D., Kirby R.C.: Deformations of spaces of imbeddings. Ann. Math. 93, 63–88 (1971) · Zbl 0214.50303 · doi:10.2307/1970753
[8] Eisenbud D., Hirsch U., Neumann W.: Transverse foliations of Seifert bundles and self homeomorphisms of the circle. Comment. Math. Helv. 56, 639–660 (1981) · Zbl 0516.57015 · doi:10.1007/BF02566232
[9] Fukui K.: Homologies of the group Diff $${\^\(\backslash\)infty(\(\backslash\)mathbb{R}\^n, 0)}$$ and its subgroups. J. Math. Kyoto Univ. 20, 475–487 (1980) · Zbl 0476.57016
[10] Hirsch M.W.: Differential Topology, Graduate Texts in Mathemetics, vol. 33. Springer, New York (1976) · Zbl 0356.57001
[11] Kotschick, D.: Stable length on stable groups, Advanced Studies in Pures Math. 52, Groups of Diffeomorphisms, pp. 401–413 (2008) · Zbl 1188.20028
[12] Lech J., Rybicki T.: Groups of C r,s -diffeomorphisms related to a foliation. Banach Center Publ. 76, 437–450 (2007) · Zbl 1122.22002 · doi:10.4064/bc76-0-21
[13] Ling W.: Translations on $${M\(\backslash\)times\(\backslash\)mathbb{R}}$$ . Amer. Math. Soc. Proc. Symp. Pure Math. 32(2), 167–180 (1978) · doi:10.1090/pspum/032.2/520533
[14] Mather J.N.: The vanishing of the homology of certain groups of homeomorphisms. Topology 10, 297–298 (1971) · Zbl 0221.57021 · doi:10.1016/0040-9383(71)90022-X
[15] Mather J.N.: Commutators of diffeomorphisms. Comment. Math. Helv. I 49, 512–528 (1974) · Zbl 0289.57014 · doi:10.1007/BF02566746
[16] Mather J.N.: Commutators of diffeomorphisms. Comment. Math. Helv.II 50, 33–40 (1975) · Zbl 0299.58008 · doi:10.1007/BF02565731
[17] Mather J.N.: Commutators of diffeomorphisms. Comment. Math. Helv. III 60, 122–124 (1985) · Zbl 0575.58011 · doi:10.1007/BF02567403
[18] McDuff D.: The lattice of normal subgroups of the group of diffeomorphisms or homeomorphisms of an open manifold. J. London Math. Soc. 18(2), 353–364 (1978) · doi:10.1112/jlms/s2-18.2.353
[19] Rybicki T.: The identity component of the leaf preserving diffeomorphism group is perfect. Monatsh. Math. 120, 289–305 (1995) · Zbl 0847.57033 · doi:10.1007/BF01294862
[20] Rybicki T.: Commutators of diffeomorphisms of a manifold with boundary. Annales Pol. Math. 68, 199–210 (1998) · Zbl 0907.57022
[21] Rybicki T.: On the group of diffeomorphisms preserving a submanifold. Demonstr. Math. 31, 103–110 (1998) · Zbl 0899.57019
[22] Schweitzer, P.A.: Normal subgroups of diffeomorphism and homeomorphism groups of $${\(\backslash\)mathbb{R}\^n}$$ and other open manifolds, preprint (2009)
[23] Thurston W.: Foliations and groups of diffeomorphisms. Bull. Amer. Math. Soc. 80, 304–307 (1974) · Zbl 0295.57014 · doi:10.1090/S0002-9904-1974-13475-0
[24] Tsuboi T. et al.: On the group of foliation preserving diffeomorphisms. In: Walczak, P. (eds) Foliations 2005, pp. 411–430. World scientific, Singapore (2006) · Zbl 1222.57031
[25] Tsuboi, T.: On the uniform perfectness of diffeomorphism groups, Advanced Studies in Pures Math. 52, Groups of Diffeomorphisms, pp. 505–524 (2008) · Zbl 1183.57024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.