# Does the image of the exponential map generate the group?

Let $G$ be a connected Fréchet-Lie group and let $\mathfrak g$ be its Lie algebra. Does the image $\exp(\mathfrak g) \subset G$ of the exponential map generate $G$?

I believe that this is open. In fact, even for the very special case where $G = \text{Diff}_0(M)$ for a smooth manifold $M$, the only proof I know that $G$ is generated by the image of the exponential map uses a very deep theorem of Thurston that says that in this case $G$ is a simple group (this implies that $G$ is generated by the image of the exponential map since the subgroup generated by the image of the exponential map is a nontrivial normal subgroup). Even giving a genuinely different proof in this special case would be very interesting.
• $Diff_0$ means the unit component of the group of $C^\infty$ diffeomorphisms with the $C^\infty$ topology? – YCor Sep 17 '17 at 21:41