Normal subgroups of $\textrm {Diff}^{\Omega }(\textbf {R}^{3})$

Abstract:

Let $\Omega$ be a volume element on ${{\mathbf {R}}^3}$ of infinite total $\Omega$-volume. We denote by ${\operatorname {Dif}}{{\text {f}}^\Omega }({{\mathbf {R}}^3})$ the group of all $\Omega$-preserving diffeomorphisms of ${{\mathbf {R}}^3}$, by ${\operatorname {Diff}}_{\text {c}}^\Omega ({{\mathbf {R}}^3})$ the subgroup of all elements with compact support and by ${\operatorname {Diff}}_{\text {f}}^{\Omega }(\mathbf {R}^3)$ the subgroup of all elements whose support has finite $\Omega$-volume. We prove that there is no normal subgroup between ${\operatorname {Diff}}_{\text {c}}^\Omega ({{\mathbf {R}}^3})$ and ${\operatorname {Diff}}_{\text {f}}^\Omega ({{\mathbf {R}}^3})$.