Example 2

Let E be a real euclidean affine space, $\mathfrak{H}$ a set of hyperplanes of E, W the group of displacements of E generated by the orthogonal reflections with respect to the hyperplanes of $\mathfrak{H}$. Assume that $\mathfrak{H}$ is stable under W and that the group W, with the discrete topology, operates properly on E.

The preceding can be applied to the operation of W on E. The fixer of a point $x$ of E is the subgroup of W generated by the reflections with respect to the hyperplanes of $\mathfrak{H}$ containing $x$ (Chap. V, §3, no. 3, Prop. 2). Consequently, the largest orbit type $\tau$ is the class of the subgroup $\{$Id$_E\}$, and $E_{(\tau)}$ is the union of the chambers of E. Note that in this case the covering $E_{(\tau)}\rightarrow E_{(\tau)}/W$ is trivial, and in particular $E_{(\tau)}$ is not connected if $\mathfrak{H}$ is non-empty.