Definitions

# Properly discontinuous action

In topology and related branches of mathematics, an action of a group G on a topological space X is called properly discontinuous if every element of X has a neighborhood that moves outside itself under the action of any group element but the trivial element. The action of the deck transformation group of a cover is an example of such action. The set of points at which G acts discontinuously is called the free regular set.

A series of related definitions, some weaker or stronger, are presented here.

## Discontinuous action

Let a group G act on a topological space X by homeomorphisms. The action of G is said to be discontinuous at $x in X$ if there is a neighborhood U of x such that the set

$\left\{gin G, |, gUcap Uneq varnothing\right\}$
is finite. The action is called discontinuous if it is discontinuous at every point.

There are several lemmas that follow from this definition, relating discontinuous action to ideas of discreteness and in particular to that of a discrete group.

Lemma: If G acts discontinuously then the orbits of the action have no accumulation points. That is, if $\left\{g_n\right\}$ is a sequence of distinct elements of G and $xin X,$ then the sequence $\left\{g_ncdot x\right\}$ has no limit points. Conversely, if X is locally compact then an action that satisfies this condition is discontinuous.

Lemma: Assume that X is a locally compact Hausdorff space and let $operatorname\left\{Aut\right\}\left(X\right)$ denote the group of self homeomorphisms of X endowed with the compact-open topology. If $rho:Gto operatorname\left\{Aut\right\}\left(X\right)$ defines a discontinuous action then the image $rho\left(G\right)$ is a discrete subset of $operatorname\left\{Aut\right\}\left(X\right).$

## Proper discontinuous action

Let a group G act on a topological space X by homeomorphisms. This action is called properly discontinuous if, for every x in X, there is a neighborhood U of x such that
$forall g in G quad \left(g neq e\right) Rightarrow \left(gU cap U = varnothing\right).$

The set U is called a nice neighborhood of x.

If X is a locally path-connected Hausdorff space, and the action of G is properly discontiuous, then the quotient map $pi:X to X/G$ is a covering map and the group of automorphisms is G itself: $operatorname\left\{Aut\right\}\left(X\right)=G.$

## Definition with a non-trivial stabilizer

The basic definition above fails for the interesting case where the stabilizer of the point x is non-trivial. The definition can be extended, as follows. Consider a subgroup $H subset G$. One then says that a set Y is precisely invariant under H in G if

$forall h in H, quad h\left(Y\right)=Y ;mbox\left\{ and \right\};$
`      forall g in G-H, quad gY cap Y = varnothing.`

Then let $G_x$ be the stabilizer of x in G. One says that G acts discontinuously at x in X if the stabilizer $G_x$ is finite and there exists a neighborhood U of x that is precisely invariant under $G_x$ in G. If G acts discontinuously at every point x in X, then one says that G acts properly discontinuously on X.

## Definition as a locally finite set

Another common definition is in terms of a locally finite set. Given any x in X, let Gx be the orbit of x under the action of G. One then says that the orbit is locally finite if every compact subset K of X contains at most a finite number of points from the orbit Gx; that is, if

$mbox\left\{card\right\} \left(Kcap Gx\right) < infty$

If the orbit Gx is locally finite for every x in X, then one says that the action of G on X is properly discontinuous.

Note that this alternate definition does not coincide with the basic definition if the stabilizer of x in G is non-trivial.