TITLE: Actions on Sⁿ with quotient Sⁿ
QUESTION [16 upvotes]: 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.

REPLY [12 votes]: 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.