What exactly is meant by the conformal group of <PERSON> space?
This is sort of a silly question because I'm a total beginner, and I debated whether it was better to ask here or on Math.SE. I decided on here because it's about how physicists use terminology, even though the terminology is intrinsically mathematical and there's not actually any physical content in the question.
It's a well-known fact in theoretical physics that the conformal group of <PERSON> space is $\operatorname{SO}(4,2)$ (or sometimes its double cover $\operatorname{SU}(2,2)$ is chosen instead). This is infinitesimally generated by the standard Poincare generators $M_{\mu \nu}$ and $P_\mu$ together with scale transformation $D$ and special conformal generators $K_\mu$. I've also seen conformal groups discussed for other manifolds (the case which lead me to this question is <PERSON>'s 2003 paper Perturbative Gauge Theory As A String Theory In Twistor Space, which includes the conformal group of $\mathbb{R}^4$ with all possible signatures.)
However, the above doesn't seem to agree with the definition of the term "conformal group" which I know from the minimal amount of conformal geometry I know. Specifically, the conformal group of a pseudo-Riemannian manifold $M$ is the group of conformal self-maps on $M$, that is, self-diffeomorphisms for which the pullback metric is conformally equivalent to the original metric. (Two metrics $g$, $h$ are conformally equivalent if there is a positive real-valued function $u$ such that at each point $x \in M$ we have $g_{x, \mu \nu}=u(x)h_{x, \mu \nu}$.)
The trouble is that the conformal group of <PERSON> space isn't $\operatorname{SO}(4,2)$ with the above definition. The special conformal transformations, given by the expression (for some vector $b_\mu$) $$ x' ^\mu = \frac{x^\mu - x^2 b^\mu}{1-2b \cdot x +b^2 x^2}$$ aren't self-maps of <PERSON> space at all. 364
The denominator can become 0 (while the numerator remains finite), and at that particular $x^\mu$ the transformation given by $b^\mu$ is undefined. For an example, take $x=(1,0,0,0)$ and $b=(0,1,0,0)$. So in fact the special conformal transformations are only maps from a subset of <PERSON> space back to the full <PERSON> space, and shouldn't be included as part of the conformal group if we take the above definition. It's difficult to tell from this formula, but from the fact that $$ \frac{x'^\mu}{x'^2} = \frac{x^\mu}{x^2} - b^\mu$$ we can tell that the inverse transformation is just given by $-b^\mu$, another special conformal transformation, which means these maps aren't surjective either.
These problems stem from the fact that the conformal group is typically derived in physics by considering the algebra of vector fields (which we think of as infinitesimal coordinate transformations) which satisfy the conformal <PERSON> equation. There's in general no reason to expect that the corresponding finite transformation to be well-behaved away from the origin.
How, exactly, are conformal groups defined in physics for arbitrary (in particular noncompact) pseudo-Riemannnian manifolds so that we get the expected result for <PERSON> space?
To be perfectly clear, I don't want answers of the form "find the Lie algebra, then exponentiate everything". I know that this works, but I want a definition for the conformal group as a group, without referencing algebras. In my opinion, there ought to be a way to define it this way, and it seems a bit important to me since a priori I don't see any reason why the group of conformal self-maps of a manifold needs to be a (possibly infinite-dimensional) Lie group at all. There are probably some high-power theorems (perhaps in PDE) which would imply this trivially, but naturally we need to know an actual group structure to be able to apply them. 364