Distal compactifications of group extensions
Let \(e\to N\to G\to K\to e\) be a short exact sequence of groups. By a result of Schreier, \(G\) is then isomorphic to \(N\times K\) with multiplication given by \((s,t)(s',t')= (st(s')[t,t'], tt')\), \(s,s'\in N,t,t'\in K\), where the Schreier mappings \(K\times K\to N\), \((t,t')\to [t,t']\) and \(N\times K\to N\), \((s,t)\to t(s)\) satisfy certain extension conditions. Suppose that \(N\) and \(K\) are topological groups and that the Schreier maps are continuous. Then \(G\) is a topological group. This paper is concerned with describing several right topological semigroup compactifications of \(G\) in terms of the corresponding compactifications of \(N\) and \(K\). As a sample, consider \(G^{LC}\), the universal right topological semigroup compactification with the property that the natural left action of \(G\) on the compactification is jointly continuous. The problem is whether \(G^{LC}\) is a right topological semigroup extension of some compactification of \(N\) with \(K^{LC}\). Simple examples show that such a result is unlikely unless \(K\) or \(N\) is compact. Suppose that \(N\) is compact. Then \(G^{LC}\) is the Schreier extension of \(N\) by \(K^{LC}\) if and only if the set \([\cdot,K]\) is equicontinuous. In particular, if \(N\) is compact and \(G\) is a semi-direct product of \(N\) and \(K\), then \(G^{LC}\) is a semi-direct product of \(N\) and \(K^{LC}\). A similar result is obtained when \(K\) is compact. The authors also investigate distal and point distal compactifications and obtain various results in the above spirit. Several illustrating examples are given. The almost periodic compactification is studied elsewhere.
distal and point distal compactifications
scientific article; zbMATH DE number 1309617