TITLE: Möbius and projective 3-manifolds
QUESTION [9 upvotes]: A projective 3-manifold is a smooth manifold that admits an atlas with values in the real projective 3-space such that all transition maps are restrictions of projective transformations. A Möbius 3-manifold is defined similarly, with the projective space replaced with the standard 3-sphere and projective transformations replaced with Möbius transformations. (Recall that a Möbius transformation of the sphere $S^n$ is a self-diffeomorphism of the sphere that preserves the angles of the standard metric; such transformations form a Lie group isomorphic to $SO_{n+1,1}(\mathbf{R}))/\pm I$).
Both projective and Möbius manifolds are particular cases of manifolds admitting an $(M,G)$-structure in the sense of W. Thurston.
Every closed (=compact, orientable and without boundary) 2-surface admits both a Möbius structure and a projective one. 
I vaguely remember having been to a talk some time ago where the speaker said that (conjecturally?) the situation in dimension 3 is similar. But I don't remember the details at all. So I would like to ask if anyone knows whether either of the statements (each closed 3-manifold admits a Möbius, resp. projective structure) is a theorem, a conjecture or becomes one or the other after eliminating some counter-examples.
A related question: if memory serves, in the same talk it was mentioned that the $PGL_{n+1}(\mathbf{R})$ and the Möbius group are (conjecturally?) the maximal groups that can act faithfully on an $n$-manifold. I was wondering if anyone knows a reference for this.

REPLY [7 votes]: The questions about manifolds admitting flat conformal or projective structures have been answered, so I'll just address your final question, which is whether $\mathrm{PGL}(n{+}1,\mathbb{R})$ and $\mathrm{O}(n{+}1,1)/\{\pm I_{n+2}\}$ are 'maximal' finite dimensional groups acting on $n$-manifolds.
The answer depends on what you mean by 'maximal'.  They are maximal in the sense that any finite dimensional Lie subgroup of $\mathrm{Diff}(\mathbb{RP}^n)$ that contains $\mathrm{PGL}(n{+}1,\mathbb{R})$ must be equal to $\mathrm{PGL}(n{+}1,\mathbb{R})$ and any finite dimensional Lie subgroup of $\mathrm{Diff}(S^n)$ that contains $\mathrm{O}(n{+}1,1)/\{\pm I_{n+2}\}$ must be equal to $\mathrm{O}(n{+}1,1)/\{\pm I_{n+2}\}$.  This is essentially due to Élie Cartan, who proved the corresponding statement that the corresponding Lie algebras of vector fields are maximal finite dimensional Lie algebras in $\mathrm{Vect}(M)$ in his series of papers on so-called 'infinite groups' of diffeomorphisms.  For a more modern treatment, see Singer and Sternberg's "The infinite groups of Lie and Cartan, Part I (the transitive case)".
However, if by maximal, you mean 'highest (finite) dimensional Lie subgroup of $\mathrm{Diff}(M^n)$', then this is false, as there is no upper bound for trivial reasons:  Take $k$ disjoint open sets $U_j\subset M$ each with compact closure, and let $X_j$ be a nonzero vector field on $M$ that is supported in $U_j$.  Then the flows of the $X_j$ commute and generate an effective action of $\mathbb{R}^k$ on $M^n$.  Since $k$ can be arbitrary, it follows that there is no upper bound on the dimension of Lie subgroups of $\mathrm{Diff}(M)$.
Of course, this example is not satisfying, and you well might ask whether there is an upper bound for transitive actions.   Even then, the answer is 'no'.  For example, for a fixed integer $m$, consider the transformations of $\mathbb{R}^2$ that are of the form
$$
\Phi(x,y) = (\ ax{+}b,\ cy + d_m x^m + d_{m-1} x^{m-1} + \cdots + d_1 x + d_0\ ) 
$$
where $a,b,c, d_0,\ldots, d_m$ are constants with $ac\not=0$.  These constitute a Lie group of dimension $m{+}4$ acting effectively and transitively on $\mathbb{R}^2$.  A similar example can obviously be constructed on $\mathbb{R}^n$ for any $n>1$.  One can construct transitive examples for any $n>1$ on compact manifolds as well.
The key missing condition necessary for classification is to ask that the group act both transitively and primitively, i.e., that the group action not preserve any nontrivial foliation of $M$.  Then there is a classification, due in large part to Élie Cartan, but finally completed to modern standards of rigor by Ochiai in the 70s.  The conformal and projective transformation groups are indeed maximal among primitive, transitive actions, but there are many others.  For example, the action of $\mathrm{PGL}(n,\mathbb{R})$ on the Grassmannian of $k$-planes in $\mathrm{R}^n$ is maximal, primitive, transitive for every pair $(k,n)$.  For a given  dimension $n$ of the underlying manifold, the projective group, at dimension $n^2{+}2n$, is the largest primitive transitive Lie subgroup in overall dimension.