# Proper actions and diffeomorphism groups

Since the diffeomorphism group is not locally compact; is it true that there is no proper action of an infinite-dimensional diffeomorphism group on a finite-dimensional smooth manifold?

Edit: The group is the $C^\infty$ self-diffeomorphisms of a finite-dimensional smooth open manifold. Let's say that the group is endowed with the Whitney $C^\infty$-topology.

• Properness will depend on the topology of the diffeomorphism group. And the whole question depends on the definition of diffeomorphism you choose (groups of $C^k$ diffeomorphisms of the circle, for instance, are very different for $k=1$, $k=2$, $k=\infty$, $k=\omega$...) – YCor Feb 20 '16 at 19:25

Let $G$ be a Hausdorff topological group acting continuously properly on a locally compact space $X$. Then $G$ is locally compact.
Indeed, fix $x\in X$ and let $Y$ be a compact neighborhood of $x$. By definition of properness, there is a compact subset $K$ of $G$ such that $g\notin K$ implies $gY\cap Y=\emptyset$. By continuity of the action, there is a neighborhood $V$ of 1 in $G$ such that $g\in V$ implies $gx\in Y$. In particular, $V\subset K$, so $K$ is a compact neighborhood of 1 in $G$. It follows that $G$ is locally compact.