  We provide a tool for studying properly discontinuous actions of non-compact
groups on locally compact, connected and paracompact spaces, by embedding such
an action in a suitable zero-dimensional compactification of the underlying
space with pleasant properties. Precisely, given such an action $(G,X)$ we
construct a zero-dimensional compactification $\mu X$ of $X$ with the
properties: (a) there exists an extension of the action on $\mu X$, (b) if $\mu
L\subseteq \mu X\setminus X$ is the set of the limit points of the orbits of
the initial action in $\mu X$, then the restricted action $(G,\mu X\setminus
\mu L)$ remains properly discontinuous, is indivisible and equicontinuous with
respect to the uniformity induced on $\mu X\setminus \mu L$ by that of $\mu X$,
and (c) $\mu X$ is the maximal among the zero-dimensional compactifications of
$X$ with these properties. Proper actions are usually embedded in the end point
compactification $\epsilon X$ of $X$, in order to obtain topological invariants
concerning the cardinality of the space of the ends of $X$, provided that $X$
has an additional "nice" property of rather local character ("property Z",
i.e., every compact subset of $X$ is contained in a compact and connected one).
If the considered space has this property, our new compactification coincides
with the end point one. On the other hand, we give an example of a space not
having the "property Z" for which our compactification is different from the
end point compactification. As an application, we show that the invariant
concerning the cardinality of the ends of $X$ holds also for a class of actions
strictly containing the properly discontinuous ones and for spaces not
necessarily having "property Z".
