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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

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
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})$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0678342-9
PII: S 0002-9947(1983)0678342-9
Article copyright: © Copyright 1983 American Mathematical Society