A characterization of totally disconnected compactly ruled groups (Q2024778)

A locally compact group \(G\) is \emph{compactly ruled} if it is a directed union of compact open subgroups. The class of compactly ruled groups is a proper subclass of the class of all compactly covered groups. Here it is shown that \(G\) is totally disconnected and compactly ruled iff the set of all compact open subgroups is dense in \(\mathrm{SUB}(G)\), where \(\mathrm{SUB}(G)\) denotes the Chabauty space of closed subgroups of \(G\). This is a natural extension of Gartside-Smith result for profinite groups. It is worth to highlight that the argument is based on the notion of a \emph{periodic element}, i.e., an element which generates a relatively compact subgroup. Notice that in literature such an element is also known as \emph{compact element}.