TITLE: How transitive are the actions of symplectomorphism groups ?
QUESTION [15 upvotes]: This question is motivated by the classical fact from differential geometry : 
Let $M$ be a smooth manifold of dimension at least $2$. Then for any $n$ the diffeomorphism group $\textrm{Diff}(M)$ acts transtively on the configuration space of $n$-points in $M$ or equivalently it acts $n$-transitively on $M$.
As I recall, it is known that the symplectomorphism group $(M,\omega)$ acts transitively on $M$, which is assumed to be symplectic. My question is then the following :
Let $(M,\omega)$ be a symplectic manifold.
(i) When does $\textrm{Symp}(M,\omega)$ act $n$-transtively for $n\geq 2$ ?
(ii) If the answer above is NOT ALWAYS then what is known ?
As some background, the usual way one proves (rather the only way I know how to prove this) the first fact is by showing the following :
(i) for two sets of distinct $n$ points in $M$ given by $\{p_1,\ldots,p_n\}$ and $\{q_1,\ldots,q_n\}$ which are close, we find disjoint disks $D_i$'s containing $p_i,q_i$. This requires dimension at least $2$. Use some diffeomorphism of $D_i$ that is smoothly identity at the boundary and looks like a rotation inside $D_i$ that swaps $p_i$ and $q_i$.
(ii) Define the natural equivalence relation on $n$-tuples and observe that the configuration space of $n$-points in $M$. By (ii) each equivalence class is open. It is alsoclosed being the complement of open sets. Since the configuration space is connected (this requires dimension at least $2$) this means there is only one equivalence class.  
Does this idea work in the symplectic setting - perhaps by taking paths $\gamma_i$ from $p_i$ to $q_i$ and getting Hamiltonian vector fields via $\omega(\gamma_i',\cdot)$ ?

REPLY [5 votes]: The main question has already been answered, but I wanted to add the following remark.  When you have a group acting transitively on a certain space then you know that the space in question will be a homogenous space that is isomorphic to the quotient of the group by the stabilizer of a point.
Every possible homogenous space is thus characterized by a closed subgroup of the group.
I read many years ago a paper by Banyaga; I think this one
A. Banyaga, Isomorphisms between classical diffeomorphism groups., Geometry, topology, and dynamics (Montreal, PQ, 1995), 1–15,
CRM Proc. Lecture Notes, 15, Amer. Math. Soc., Providence, RI, 1998.
Here he characterized several types of geometric structures by the isomorphism groups.  The question was for example:  Suppose that you know that the group of symplectomorphisms of two symplectic manifolds are isomorphic, can you then reconstruct or identify the underlying symplectic manifolds.  As far as I remember, he characterized several structures like smooth manifold, symplectic structures, contact structures, volume forms etc. where knowledge of the isomorphism group (forgetting topology and only remembering the algebraic structure) allows you to identify the underlying geometric structure (in the symplectic case up to rescaling).
For a certain isomorphism group to have such a property, he needed the group to act n-transitively (realized by an isotopy with arbitrarily small support) for any n.  I do not remember the details anymore, but the result was somehow that the stabilizer of any point in such a group is a maximal subgroup and if we have a group isomorphism between two groups mapping one maximal subgroup to another one, then the isomorphism induces an identification between the underlying geometric space.
I apologize for not remembering more details, but I found the result quite interesting at time I read this.