# Action of discrete subgroups E(n) on $\Bbb{R}^n$

Isometry group of euclidean space $\Bbb{R}^n$ is displayed by E(n). We say that a subgroup G of E(n) is discrete if and only if the subspace topology (from E(n)) on G is discrete.

If X and Y are Hausdorff spaces and Y is locally compact, then continuous map $f:X\longrightarrow Y$ is called proper if and only if for each compact subset $K\subseteq Y$, the pre-image i.e $f^{-1}(K)$ is compact.

An continuous action of an arbitrary topological group G on the space X is called proper if the associated map $G \times X\longrightarrow X\times X$ that $(g,x)\mapsto (gx, x)$ is proper.

My question:

Let G to be discrete subgroup of E(n). Is any (continuous)action of G on $\Bbb{R}^n$ proper?

As a counterexample, you may consider an irrational rotation on $\mathbb{R}^2$.
• Rotation group on $\Bbb{R}^2$ is SO(2) that is not discrete. – bigli Nov 29 '13 at 8:12
• Why is action of the subgroup generated by a single irrational rotation on $\Bbb{R}^2$, proper? – bigli Nov 29 '13 at 10:57