Normal subgroups of $\textrm {Diff}^{\Omega }(\textbf {R}^{n})$

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 ${\text {Diff}}_f^\Omega ({{\mathbf {R}}^n})$.