Annotated Definitions

Proper discontinuity

(Credit where due: this is a summary things from MathOverflow 55726 and Math Stack Exchange 1082834.)

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 KK \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.

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