Let $\Gamma$ be a discrete subgroup of $R^n \rtimes O(n)$. Let $X(\Gamma)$ be the set of reflections in $\Gamma$(i.e. the elements that fix some hyperplane in $R^n$.) and let $\{H_\alpha\}_\alpha$ be the set of reflection hyperplanes.
I was trying to show for myself that any element of $\Gamma$ takes a connected component of $R^n - \cup_\alpha H_\alpha$ to a connected component.
My proof rests on the fact that $\Gamma$ acts on $R^n - \cup_\alpha > H_\alpha$. Why is this true?
What I know so far/motivation: I declare the distance between two elements in $\Gamma$ is the magnitude of the distance of the translation components of the elements + the distance in the operator norm between the $O(n)$ components. Thus up to a constant, the distance between two reflections is difference between the normals from $(0,...0)$ to the hyperplanes + the magnitude of the angle between the two hyperplanes. Thus any two hyperplanes must either have a minimum angle between them or the difference between the normals from $(0,..0)$ to the hyperplanes must be of a minimum magnitude.
My end goal is to show that the reflections corresponding to the hyperplanes bounding a connected component $R^n - \cup_\alpha H_\alpha$ generate $\Gamma$, and I hope to do this by showing that each connected component is a fundamental domain for the action of $\Gamma$ on $R^n- \cup_\alpha H_\alpha$, and I think all of my steps above are necessary for me to prove this. (Once I know that $\Gamma$ acts on $R^n - \cup_\alpha H_\alpha$, I know of a distance argument that shows that this is indeed the case.)