# How transitive are the actions of symplectomorphism groups ?

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 $\omega(\gamma_i',\cdot)$ ?

• Did you mean n-transitively on the configuration space of n-points, or n-transitively on M? It seems like your proof addresses the later, but you said the former. – Joey Hirsh Apr 17 '11 at 5:41
• @ Joey - I edited the question now. You're right that I meant $n$-transitivity on $M$ and this is equivalent to the usual transitivity on configuration space of $n$ points in $M$. – Somnath Basu Apr 17 '11 at 5:45

Dear Somnath Basu, the answer to question is that, for any $k\geq 2$, the k-fold transitivity of the action of $\mathrm{Sympl}(M,\omega)$ on $M$ has only one obstruction, the trivial one, i.e. connectivity of $M$.
But has you proposed there is even more.

In particular Theorem A in a paper of W. Boothby says that,

given a connected symplectic manifold $(M,\omega)$, for any two sets $\{x_1,\ldots,x_n\}$ and $\{y_1,\ldots,y_n\}$ of disjoint point in $M$, where $n$ is arbitrary natural number, there exists a time dependent hamiltonian vector field $X_t$ of $(M,\omega)$ such that its evolution operator $K^X_{1,0}$ maps $x_i$ to $y_i$ for $i=1,\ldots,n$.

• That's what I surmised! Thanks for the reference. – Somnath Basu Apr 17 '11 at 14:29

For any finite number of points, the subgroup of Hamiltonian symplectomorphism that fixes some neighbourhood of the points acts transitively on the complement of the points.

Choose two points $a, b$ in the complement, and a path avoiding the constrained points joining them. Then you can find a (non-autonomous) Hamiltonian function with support in an arbitrarily small neighborhood of the path whose time-1 flow maps $a$ to $b$. Indeed, the group of Hamiltonian tranformations with support in a connected neighbourhood is transitive on Darboux balls charts, so the orbit of a point is open and closed.

What this argument seems to prove is in fact the statement :

For any connected open subset $\Omega$ in a symplectic manifold, the group of Hamiltonian symplectomorphisms with support in $\Omega$ is transitive in $\Omega$.

In particular, the group of compactly Hamiltonian symplectomorphisms is $n$-transitive $\forall n$ on any connected symplectic manifold.

Very late, let me point out the following paper:

Peter W. Michor, Cornelia Vizman: n-transitivity of certain diffeomorphism groups. Acta Math. Univ. Comenianiae 63, 2 (1994),221--225. arXiv:dg-ga/9406005 (pdf)

There $n$-transitivity is proved for many groups of diffeomorphisms, in particular also for the groups of real analytic symplectic, or volume preserving, or contact diffeomorphisms.

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.