ON EMBEDDINGS OF PROPER AND EQUICONTINUOUS ACTIONS IN ZERO-DIMENSIONAL COMPACTIFICATIONS

Abstract. 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 compactifica-tion of the underlying space with pleasant properties. Precisely, given such an action (G,X) we construct a zero-dimensional compactification µX of X with the properties: (a) there exists an extension of the action on µX, (b) if µL ⊆ µX \ X is the set of the limit points of the orbits of the initial action in µX, then the restricted action (G,µX \ µL) remains properly discontinu-ous, is indivisible and equicontinuous with respect to the uniformity induced on µX \ µL by that of µX, and (c) µX is the maximal among the zero-dimensional compactifications of X with these properties. Proper actions are usually embedded in the end point compactification εX of X, in order to ob-tain topological invariants concerning the cardinality of the space of the ends of X, provided that X has an additional “nice ” property of rather local char-acter (“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 com-pactification 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”