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...



Finite-dimensional subgroups of circle diffeomorphism group
If $G$ is a connected Lie group acting transitively and faithfully on a connected smooth $1$-manifold, then $G$ is at most $3$-dimensional; in fact its Lie algebra embeds in that of $SL\_2(\mathbb R)$. (Edit: Alex Eskin's answer says this in detail, with a reference.)


Each orbit of an action of a connected topological group on a $1$-manifold is either open or one point. Thus if $G\_i$ has only one non-fixed orbit then $G\_i$ is at most $3$-dimensional.


Let $F\_i\subset S^1$ be the set of points fixed by the action of $G\_i$. The intersection of all the $F\_i$ is the set of points fixed by the union of the $G\_i$, so surely it is empty if the union of the $G\_i$ is dense. By compactness it follows that for all big enough $i$ $F\_i$ is empty, making $G\_i$ at most $3$-dimensional.


