Minimality of the Semidirect Product

Abstract

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. We provide a sufficient and necessary condition for the minimality of the semidirect product $G⋋P,$ where $G$ is a compact topological group and $P$ is a topological subgroup of $Aut(G)$. We prove that $G⋋P$ is minimal for every closed subgroup $P$ of $Aut(G)$. In case $G$ is abelian, the same is true for every subgroup $P\subseteq Aut(G)$. We show, in contrast, that there exist a compact two-step nilpotent group $G$ and a subgroup $P$ of $Aut(G)$ such that $G⋋P$ is not minimal. This answers a question of Dikranjan. Some of our results were inspired by a work of Gamarnik.

16 pages. arXiv admin note: substantial text overlap with arXiv:1501.03410

References in corpus (1)

• The maximal G-compactifications of G-spaces with special actions