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 compacti. cation of the underlying space with pleasant properties. Precisely, given such an action (G, X) we construct a zero-dimensional compacti. cation mu X of X with the properties: (a) there exists an extension of the action on mu X, (b) if mu L subset of mu X \ 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 \ mu L) remains properly discontinuous, is indivisible and equicontinuous with respect to the uniformity induced on mu X \ mu L by that of mu X, and (c) mu X is the maximal among the zerodimensional compacti. cations of X with these properties. Proper actions are usually embedded in the endpoint compacti. cation 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 compacti. cation coincides with the endpoint one. On the other hand, we give an example of a space not having the "property Z" for which our compacti. cation is different from the endpoint compacti. cation. 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".
