A finite subgroup $G$ of the orthogonal group acts on the unit sphere $\mathbb{S}^{n-1}$ in $\mathbb{R}^n$, and we can ask how the topology of the quotient $\mathbb{S}^{n-1}/G$ is related to the structure of $G$. The first question in this direction — for what $G$ is the quotient a sphere? — has recently been answered by Christian Lange. The proof of his result is a delicate case-based induction. A natural followup question is to find a uniform proof. One possibility is to impose a $G$-invariant cell complex structure on $\mathbb{S}^{n-1}$ and then to utilize tools from discrete geometry such as shellings and discrete Morse theory. In this talk I will present an outline of Lange's result and some notes toward this followup program, focusing on subgroups of Coxeter groups.
My interest in the problem actually comes from the invariant theory of finite groups, although this motivation will be discussed only in passing in the talk.