Actions on Sⁿ with quotient Sⁿ 
What is known about isometric actions on $\mathbb S^n$ such that the quotient space is homeomorphic to $\mathbb S^n$?

Comments.


*

*I am mostly interested in (maybe trivial) properties of such actions for large $n$.

*I see that the orientation preserving part of Coxeter's group has this property. 

*Originally I thought that any such action is generated by rotations around $\mathbb S^{n-2}$'s; now I see that there are other examples for $\mathbb S^3$; thanks to Lee Mosher.

 A: In the following article of M.A.Mikhailova (М.А. Михайлова)
Изв. АН СССР. Сер. матем., 48:1 (1984)
О ФАКТОРПРОСТРАНСТВЕ ПО ДЕЙСТВИЮ КОНЕЧНОЙ ГРУППЫ, ПОРОЖДЕННОЙ ПСЕВДООТРАЖЕНИЯМИ.
http://www.mathnet.ru/links/33220b8c84645bec685e85bf17d65994/im1420.pdf
it is proven:
Theorem. The quotient $\mathbb R^n/G$ by a linear action of a finite group $G$ is homeomorphic to $\mathbb R^n$ if and only if $G$ is generated by pseudo-reflections (i.e, rotations of $\mathbb R^n$ that fix a subspace of codimension 2). 
The proof relies on a complete classification of finite groups generated by pseudo-reflections (there is a reference to this classification at the end of the article)
(there should be of course an English translation of this article, but I can not find it now).
A: Your question translates into the language of orbifolds as saying: what is known about spherical $n$-orbifolds with underlying space homeomorphic to $S^n$? 
In $S^2$, the examples you give are all there are. 
Orbifolds with the geometry of $S^3$ were enumerated by William Dunbar in his thesis. His published paper MR1118824 contains the enumeration of the 21 oriented $S^3$-orbifolds which do not have a circle fibration over a 2-orbifold. The equivalence relation here is up to orientation preserving isometry; if you allow orientation reversing isometry then the list is cut down somewhat. Each of the 21 has underlying space homeomorphic to $S^3$. At the end of Dunbar's paper you will see that exactly 8 of the 21 are orientable double covers of Coxeter group quotients, with the corresponding Dynkin diagrams listed out explicitly. That leaves 13 examples as you ask for in $S^3$. 
A: I think I have completely answered the question in the following form:
Theorem. For a finite subgroup $\Gamma < O(n)$ the quotient space $S^{n-1}/\Gamma$ is homeomorphic to $S^{n-1}$ if and only if $\Gamma$ has the form
\begin{eqnarray*}
  \Gamma = \Gamma_{ps} \times P_1 \times \ldots \times P_k
\end{eqnarray*}
for a pseudoreflection group $\Gamma_{ps}$ and Poincaré groups $P_i<SO(4)$, $i=1,\ldots,k$, such that the factors act in pairwise orthogonal spaces and such that $n>5$ if $k=1$. 
cf. http://arxiv.org/abs/1307.4875.
Here, a pseudoreflection group is understood in the sense of Mikhailova and a Poincaré $P$ group comes from the binary icosahedral group in $SU(2)$, i.e. $S^3/P$ is Poincaré's homology sphere. 
