On embeddings of proper and equicontinuous actions in zero-dimensional compactifications

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 endpoint compactification $\varepsilon X$ 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 endpoint 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 endpoint 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".