This question is a generalization of my previous question about the circle to arbitrary manifolds.

Is there a smooth manifold M with the following property.

There exists a sequence of connected finite-dimensional subgroups Gi of M's diffeomorphism group G such that

(a) Gi is contained in Gj for i < j

(b) The union of Gi is dense in G

To remove doubt, "finite dimensional subgroup of M's diffeomorphism group" means a Lie group H with smooth faithful action on M.

The answer to my previous question established that S1 is not such a manifold. I suspect that the answer to the general question is still "no". However, the proof would have to be more sophisticated since in the case of S1 we had essentially a closed list of possible "Hs".

There is another closely related question. Fix a smooth manifold M. Consider connected Lie groups H with faithful and transitive smooth action on M. Is there an upper bound of H's dimension? For S1 the answer was "yes, 3".

This question is a generalization of my previous question about the circle to arbitrary manifolds.

Is there a smooth manifold M with the following property.

There exists a sequence of connected finite-dimensional subgroups Gi of M's diffeomorphism group G such that

(a) Gi is contained in Gj for i < j

(b) The union of Gi is dense in G

To remove doubt, "finite dimensional subgroup of M's diffeomorphism group" means a Lie group H with smooth faithful action on M.

The answer to my previous question established that S1 is not such a manifold. I suspect that the answer to the general question is still "no". However, the proof would have to be more sophisticated since in the case of S1 we had essentially a closed list of possible "Hs".

There is another closely related question. Fix a smooth manifold M. Consider connected Lie groups H with faithful and transitive smooth action on M. Is there an upper bound of H's dimension? For S1 the answer was "yes, 3".

This question is a generalization of my previous question about the circle to arbitrary manifolds.

Is there a smooth manifold M with the following property.

There exists a sequence of connected finite-dimensional subgroups Gi of M's diffeomorphism group G such that

(a) Gi is contained in Gj for i < j

(b) The union of Gi is dense in G

To remove doubt, "finite dimensional subgroup of M's diffeomorphism group" means a Lie group H with smooth faithful action on M.

The answer to my previous question established that S1 is not such a manifold. I suspect that the answer to the general question is still "no". However, the proof would have to be more sophisticated since in the case of the circle we had essentially a closed list of possible "Hs".

This question is a generalization of my previous question about the circle to arbitrary manifolds.

Is there a smooth manifold M with the following property.

There exists a sequence of connected finite-dimensional subgroups Gi of M's diffeomorphism group G such that

(a) Gi is contained in Gj for i < j

(b) The union of Gi is dense in G

To remove doubt, "finite dimensional subgroup of M's diffeomorphism group" means a Lie group H with smooth faithful action on M.

The answer to my previous question established that S1 is not such a manifold. I suspect that the answer to the general question is still "no". However, the proof would have to be more sophisticated since in the case of S1 we had essentially a closed list of possible "Hs".

There is another closely related question. Fix a smooth manifold M. Consider connected Lie groups H with faithful and transitive smooth action on M. Is there an upper bound of H's dimension? For S1 the answer was "yes, 3".