Free and properly discontinuous group actions

Let $G$ be a group acting on a topological space $X$ by homeomorphisms, so that for each $g\in G$ the map $x\mapsto g\cdot x$ is a homeomorphism of $X$ (Left group actions, transitive actions, and faithful actions, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological). The action is free when $g\cdot x=x\text{for some }x\in X⟹g=e,$ that is, when no nonidentity element of $G$ fixes a point (A free group action has no nonidentity element fixing a point). The action is properly discontinuous when for every compact subset $K\subseteq X$ (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right) the set of group elements that move $K$ to meet itself, $\{g\in G:g\cdot K\cap K\mathrm{\ne }\mathrm{\varnothing }\},$ is finite. A group $G$ acting freely and properly discontinuously means that both conditions hold; the two are independent in general. The same names are used when $G$ is given as a subgroup of the homeomorphism group of $X$ acting by evaluation.

For the plane, a subgroup $\mathrm{\Lambda }\le (\mathbb{C},+)$ acts on $\mathbb{C}$ by the translations $z\mapsto z+\lambda$, which are biholomorphisms. Such a $\mathrm{\Lambda }$ is a plane lattice of rank two, or a rank-two lattice, when $\mathrm{\Lambda }=\mathbb{Z}v+\mathbb{Z}w=\{mv+nw:m,n\in \mathbb{Z}\}$ for some $v,w\in \mathbb{C}$ that are linearly independent over $\mathbb{R}$; the pair $v,w$ is then a basis of the lattice. A subgroup $\mathrm{\Lambda }\le (\mathbb{C},+)$ is discrete when every point of $\mathbb{C}$ has a neighbourhood meeting $\mathrm{\Lambda }$ in at most one point. Whether a given translation group is discrete, free or properly discontinuous is a property of the group, not part of this terminology, and is verified in the results that use it.

Finitely many translates meet a set contained in a compact set. Let the action of $G$ on $X$ be properly discontinuous, let $K\subseteq X$ be compact and let $V\subseteq K$. If $g\cdot V\cap V\mathrm{\ne }\mathrm{\varnothing }$ for some $g\in G$, then $g\cdot K\cap K\supseteq g\cdot V\cap V\mathrm{\ne }\mathrm{\varnothing }$, so $g$ lies in the finite set $\{g\in G:g\cdot K\cap K\mathrm{\ne }\mathrm{\varnothing }\}$ of the definition. Hence $g\cdot V\cap V=\mathrm{\varnothing }$ for every $g$ outside that finite set. In particular, on a locally compact space every point has a compact neighbourhood $K$ with interior $V$, and only finitely many group elements map $V$ to meet $V$. As a second special case, if $G$ is finite then the action is automatically properly discontinuous, the displayed set being contained in the finite group.