Exercise 18, § 2

Let $G$ be a Hausdorff topological group operating continuously on a Hausdorff topological space $X$. $G$ is said to be proper at a point $x_0 \in X$ if the orbit $G.x_0$ is closed in $X$ and if $G$ operates properly on $G.x_0$ (Chapter III, § 4, no. 1).

a) Let $G$ be a locally compact topological group operating continuously on a Hausdorff uniform space $X$. Suppose that the set of homeomorphisms $x \to s.x$ of $X$, as $s$ runs through $G$, is equicontinuous. Show that if $G$ is proper at a point $x_0 \in X$, then there is a neighbourhood $V$ of $x_0$ such that the set of elements $s \in G$ such that $s.V \cap V \neq \varnothing$ is relatively compact in $G$ (cf. Chapter III, § 4, no. 4, Proposition 7). Deduce that the set $D$ of points of $X$ at which $G$ is proper is open in $X$ and that $G$ operates properly on $D$. If in addition $X$ is locally compact, show that $D$ is both open and closed in $X$ (use no. 1, Proposition 1, Corollary 2).

b) In the real plane $\mathbf{R}^2$, let $E$ be the set consisting of the origin $(0, 0)$ and the points $(0, 2^{-n})$ as $n$ runs through the integers $\geqslant 0$. For each $n \in \mathbf{Z}$ such that $n \neq 0$ let $u_n$ be the restriction to $E$ of an affine linear mapping of $\mathbf{R}^2$ into itself, such that $u_n(0, 0) = (n, 0)$, and $u_n(0, 2^{-|n|}) = (0, \theta^n)$ where $\theta$ is an irrational number such that $0 < \theta < 1$. Let $X$ be the (locally compact) subspace of $\mathbf{R}^2$ which is the union of $E$ and the sets $u_n(E)$ for $n \in \mathbf{Z}, n \neq 0$. Furthermore, let $u_0$ denote the identity mapping of $E$, and extend $u_n$ ($n \in \mathbf{Z}$) to the whole of $X$ by putting $u_n(u_m(x)) = u_{n+m}(x)$ for all $x \in E$ and all $m \in \mathbf{Z}$. The set $G$ of mappings $u_n$ is a group of homeomorphisms of $X$ which is given a discrete topology so that $G$ operates continuously on $X$. Show that $G$ is proper at every point of $X$, but is not equicontinuous and does not operate properly on $X$.

c) Let $X$ be the (not locally compact) subspace of $\mathbf{R}^2$ (identified with $\mathbf{C}$) which is the union of the half-plane $y > 0$ and the origin. Define a homeomorphism $u$ of $X$ onto itself by putting $u(0, 0) = (0, 0)$ and $u(re^{i\omega}) = re^{i\omega'}$ where $\omega' = \frac{1}{2}\omega$ if $0 < \omega \leqslant \frac{1}{2}\pi$, $\omega' = \omega - \frac{1}{4}\pi$ for $\frac{1}{2}\pi \leqslant \omega \leqslant \frac{3}{4}\pi$, $\omega' = 2\omega - \pi$ for $\frac{3}{4}\pi \leqslant \omega < \pi$ ($r < 0, 0 < \omega < \pi$). Let $G$ be the group of homeomorphisms of $X$ generated by $u$, and endow $G$ with the discrete topology. Show that $G$ is equicontinuous on $X$ and that the set of points of $X$ at which $G$ is proper is the half-plane $y > 0$.