Abstract: For a group action $G \curvearrowright X$, the canonical quotient map $\pi \colon X \to X/G$ does not admit a section in general. It is well-known that when $G \subset O(n)$ is generated by reflections, the map $\pi$ admits a section. In this lecture, we will show that the converse is also true. More generally, if $G$ is a properly discontinuous subgroup of the Euclidean group and $\pi$ admits a section, then $G$ is generated by affine reflections.