TITLE: The group of diffeomorphisms with compact support
QUESTION [7 upvotes]: Let $M$ be a topological/differentiable manifold. Is there any topology on the group of homeomorphisms/diffeomorphisms with compact support, turning it into a (locally-)compact topological group?
(My question is motivated by the fact that the isometries of a (locally-)compact metric space with finitely many connected components form a (locally-)compact topological group, and I'd like to see how far this analogy can be pushed.)

REPLY [7 votes]: The group $\mathrm{Diff}_c(M)$ is in a natural way a Lie group modeled on nuclear (LF) spaces (like the space of test functions). This was shown in 

Peter W. Michor: Manifolds of smooth maps II: The Lie group of diffeomorphisms of a non compact smooth manifold. Cahiers Topologie Geometrie Differentielle 21 (1980), 63--86.

The connected component of this group is simple (Thurston).
To the answer of Francois Ziegler one can add the following theorem of Omori:
If a Banach Lie group $G$ acts faithfully on a compact manifold (or a noncompact finite dimensional one via diffeomorphisms with compact support), then $G$ is finite dimensional.