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It's known that if a compact Lie group $G$ acts freely on a compact manifold $M$, then the orbit space $M/G$ is a manifold. If we only assume that $G$ acts almost freely (i.e. $G_x$ is finite for any $x\in M$ and there are only finitely many $x$ such that $G_x$ is not trivial),then can we deduce that $M/G$ is a orbifold or even a good orbifold (i.e. the universal covering is a manifold)?

And is there any reference for such kind of actions?

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  • $\begingroup$ I don't understand the question. Is the point that you are not insisting that the action of $G_x$ on a neighbourhood of $x$ be locally linear? If the action is locally linear, then surely $M/G$ is an orbifold by definition and the orbifold universal covering of $M/G$ is the ordinary universal covering of $M$. $\endgroup$
    – IJL
    Jan 14 at 10:04
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    $\begingroup$ I think your definition of almost free is not standard. (I don't think there should be a condition on the amount of $x$'s with a non trivial stabilizer. $\endgroup$
    – Thomas Rot
    Jan 14 at 12:54

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I think the answer to the first question is yes and the answer to the second one is no:

Yes, the quotient is an orbifold. The action of the finite group $G_x$ in a neighbourhood of $x$ can be linearized (at least if the action is by diffeomorphisms, I don't know about $C^0$ regularity), and the quotient $M/G$ is locally modelled on $G_x \backslash T_xM / T_x (G\cdot x)$.

No this orbifold is not good in general. For instance, you can glue a solid torus with a trivial circle fibration to a solid torus with a Seifert fibration with one singular fiber in the center and get a closed Seifert 3-manifold with one singular fiber. The fibration is given by the orbits of an action of S^1 and the quotient orbifold is a sphere with a single orbifold point, the simplest example of an orbifold not covered by a manifold.

More generally, I think every orbifold $M$ if dimension $n$ is the quotient of a manifold $P$ by an almost free action of the orthogonal group $O_n$ ($P$ is the principal $O_n$-bundle associated with the orbifold tangent bundle equipped with an orbifold Riemannian metric).

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