Is there a sequence of connected finite-dimensional subgroups Gi of the circle diffeomorphism group G with the following properities:
(a) Gi is contained in Gj for i < j
(b) The union of Gi is dense in G
More rigorously "finite dimensional subgroup of circle diffeomorphism group" means a Lie group H with smooth faithful action on the circle.
In order to make sense of property (b) I have to specify a topology on G. I suspect that all reasonable topologies will yield the same answer, but for the sake of definiteness let's use the "sup-norm" topology. That is, given two diffeomorphism g1 and g2, I define the distance d(g1, g2) as
supremum over x in S1 of d(g1(x), g2(x))
Here the latter d is the usual distance on the circle. This is a metric and it induces a topology.
I suspect that the answer to my question is "no". Moreover, I suspect that there is no H as above with dimension > 3. But I might be wrong...