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 compactification 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 $μL\subseteq μX\setminus X$ is the set of the limit points of the orbits of the initial action in $μX$, then the restricted action $(G,μX\setminus μL)$ remains properly discontinuous, is indivisible and equicontinuous with respect to the uniformity induced on $μX\setminus μL$ 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 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".