Annotated Definitions

# Proper discontinuity

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Let G be a locally-compact group acting on a locally-compact Hausdorff topological space X (these additional conditions on G and X being certainly satisfied if e.g. G is finitely-generated and X is a proper metric space.)

We say the action is properly discontinuous if for any compact set $K \subset X$, $K \cap gK \neq \varnothing$ for at most finitely many $g \in G$. (cf. the action of $\mathbb{Z}^n$ on $\mathbb{R}^n$ by translations, which is discontinuous: for any [sufficiently small] compact set K$K \cap gK = \varnothing$ for every $g \in \mathbb{Z}^n$; here we allow finite overlap even at arbitrarily small resolution: this allows for finite stabilizers, parabolic elements, etc.)

In particular this implies that every point $x \in X$ has an open neighborhood U s.t. $U \cap gU \neq \varnothing$ for at most finitely many $g\ in G$. (Take U to sit inside some compact set K.)

Note that the second condition is strictly weaker than the first: consider the $\mathbb{Z}$-action on the punctured plane given by $n \cdot (x,y) = (2^n x, 2^{-n}y)$. This satisfies the second condition, but not the first—take K to be any closed annulus centered about the origin, for instance.

The main point of imposing these mildly technical conditions when we study “geometric” group actions is this:

This second condition is equivalent to the quotient map $X \to X/G$ being a covering map. The first condition (the definition) serves to ensure that the map $G \times X \to X \times X$ given by $(g,x) \mapsto (x,gx)$ is proper, and hence that the quotient X/G is Hausdorff.