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Transactions of the American Mathematical Society

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Normal subgroups of $ {\rm Diff}\sp{\Omega }({\bf R}\sp{n})$


Author: Francisca Mascaró
Journal: Trans. Amer. Math. Soc. 275 (1983), 163-173
MSC: Primary 58D05; Secondary 57R50
DOI: https://doi.org/10.1090/S0002-9947-1983-0678342-9
MathSciNet review: 678342
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Abstract: Let $ \Omega $ be a volume element on $ {{\mathbf{R}}^n}$. $ {\text{Dif}}{{\text{f}}^\Omega }({{\mathbf{R}}^n})$ is the group of $ \Omega $-preserving diffeomorphisms of $ {{\mathbf{R}}^n}$. $ {\text{Diff}}_W^\Omega ({{\mathbf{R}}^n})$ is the subgroup of all elements whose set of nonfixed points has finite $ \Omega $-volume. $ {\text{Diff}}_f^\Omega ({{\mathbf{R}}^n})$ is the subgroup of all elements whose support has finite $ \Omega $-volume. $ {\text{Diff}}_c^\Omega ({{\mathbf{R}}^n})$ is the subgroup of all elements with compact support. $ {\text{Diff}}_{{\text{co}}}^\Omega ({{\mathbf{R}}^n})$ is the subgroup of all elements compactly $ \Omega $-isotopic to the identity.

We prove, in the case $ {\text{vo}}{{\text{l}}_{\Omega }}{{\mathbf{R}}^n} < \infty $ and for $ {\text{n}} \geqslant {\text{3}}$ that any subgroup of $ {\text{Dif}}{{\text{f}}^\Omega }({{\mathbf{R}}^n})$, $ N$, is normal if and only if $ {\text{Diff}}_{{\text{co}}}^\Omega ({{\mathbf{R}}^n}) \subset N \subset {\text{Diff}}_c^\Omega ({{\mathbf{R}}^n})$. If $ {\text{vo}}{{\text{l}}_{\Omega }}{{\mathbf{R}}^n} = \infty $, any subgroup of $ {\text{Dif}}{{\text{f}}^\Omega }({{\mathbf{R}}^n})$, $ N$, satisfying $ {\text{Diff}}_{{\text{co}}}^\Omega ({{\mathbf{R}}^n}) \subset N \subset {\text{Diff}}_c^\Omega ({{\mathbf{R}}^n})$ is normal, for $ n \geqslant {\text{3}}$, there are no normal subgroups between $ {\text{Diff}}_W^\Omega ({{\mathbf{R}}^n})$ and $ {\text{Dif}}{{\text{f}}^\Omega }({{\mathbf{R}}^n})$ and for $ n \geqslant 4$ there are no normal subgroups between $ {\text{Diff}}_c^\Omega ({{\mathbf{R}}^n})$ and $ {\text{Diff}}_f^\Omega ({{\mathbf{R}}^n})$.


References [Enhancements On Off] (What's this?)

  • [1] A. Banyaga, Sur la structure du groupe des diffeomorphismes qui préservent une forme symplectique, Comment. Math. Helv. 53 (1978), 174-227. MR 490874 (80c:58005)
  • [2] M. Berger, P. Gauduchon and E. Mazet, Le spectre d'une variété Riemanniene, Lecture Notes in Math., vol. 194, Springer-Verlag, Berlin and New York, 1971. MR 0282313 (43:8025)
  • [3] D. Burgelea and R. Lashof, The homotopy type of the space of diffeomorphisms. I, II, Trans. Amer. Math. Soc. 196 (1974), 1-50. MR 0356103 (50:8574)
  • [4] J. Cerf, Sur les diffeomorphismes de la sphere de dimension trois $ (\Gamma_{4} = 0)$, Lecture Notes in Math., vol. 53, Springer-Verlag, Berlin and New York, 1968. MR 0229250 (37:4824)
  • [5] D. B. A. Epstein, The simplicity of certain groups of homeomorphisms, Compositio Math. 22 (1970), 165-173. MR 0267589 (42:2491)
  • [6] R. E. Greene and K. Shiohama, Diffeomorphisms and volume preserving embeddings of noncompact manifolds, Trans. Amer. Math. Soc. 255 (1979), 403-414. MR 542888 (80k:58031)
  • [7] A. B. Krygin, Continuation of diffeomorphisms preserving volume, Functional Anal. Appl. 5 (1971), 147-150. MR 0368067 (51:4309)
  • [8] W. Ling, Simple and perfect groups of manifold automorphism germs, preprint.
  • [9] -, Real analytic diffeomorphisms of $ {R^n}$. I, preprint.
  • [10] F. Mascaro, On the normal subgroups of the group of volume preserving diffeomorphisms of $ {{\mathbf{R}}^n}$ for $ n \geqslant {\text{3}}$, Thesis, University of Warwick.
  • [11] J. Mather, Notes on topological stability, Harvard University, 1970.
  • [12] D. McDuff, The lattice of normal subgroups of the group of diffeomorphisms or homeomorphisms of an open manifold, J. London Math. Soc. (2) 18 (1978), 353-364. MR 509952 (80c:58006)
  • [13] -, On the group of volume preserving diffeomorphisms of $ {{\mathbf{R}}^n}$, Trans. Amer. Math. Soc. 261 (1980), 103-113. MR 576866 (81k:58019)
  • [14] -, On the group of volume preserving diffeomorphisms and foliations with transverse volume form, Proc. London Math. Soc. 43 (1981), 295-320. MR 628279 (83g:58007)
  • [15] -, On tangle complexes and volume preserving diffeomorphisms of open $ 3$-manifolds, Proc. London Math. Soc. 43 (1981), 321-333. MR 628280 (83g:58008)
  • [16] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294. MR 0182927 (32:409)
  • [17] R. Narasimhan, Analysis on real and complex manifolds, Advanced Studies in Pure Math., North-Holland, Amsterdam, 1968. MR 0251745 (40:4972)
  • [18] R. S. Palais, Local triviality of the restriction map for embedding, Comment. Math. Helv. 34 (1960), 305-312. MR 0123338 (23:A666)
  • [19] S. Sacks, Theory of the integral, Dover, New York, 1964. MR 0167578 (29:4850)
  • [20] W. Thurston, On the structure of the group of volume preserving diffeomorphisms (to appear).

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DOI: https://doi.org/10.1090/S0002-9947-1983-0678342-9
Article copyright: © Copyright 1983 American Mathematical Society

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