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 $G$?



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


