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 $\mathbb{R}^2$.