Locally compact groups and locally minimal group topologies
Volume 244 / 2019
Abstract
Minimal groups are Hausdorff topological groups $G$ satisfying the open mapping theorem with respect to continuous isomorphisms, i.e., every continuous isomorphism $G\to H$, with $H$ a Hausdorff topological group, is a topological isomorphism. A topological group $(G, \tau )$ is called locally minimal if there exists a neighbourhood $V$ of the identity such that for every Hausdorff group topology $\sigma \leq \tau $ with $V \in \sigma $ one has $\sigma = \tau $. Minimal groups, as well as locally compact groups, are locally minimal. According to a well known theorem of Prodanov, every subgroup of an infinite compact abelian group $K$ is minimal if and only if $K$ is isomorphic to the group $\mathbb {Z}_{p}$ of $p$-adic integers for some prime $p$.
We find a remarkable connection of local minimality to Lie groups and $p$-adic numbers by means of the following results extending Prodanov’s theorem: every subgroup of a locally compact abelian group $K$ is locally minimal if and only if either $K$ is a Lie group, or $K$ has an open subgroup isomorphic to $\mathbb {Z}_{p}$ for some prime $p$. In the nonabelian case we prove that all subgroups of a connected locally compact group are locally minimal if and only if $K$ is a Lie group, resolving Problem 7.49 from Dikranjan and Megrelishvili (2014) in the positive.
