Key subgroups and co-minimality in topological groups

Michael Megrelishvili, Menachem Shlossberg

Published 2023-09-13Version 1

We introduce a minimality property for subgroups of topological groups. A subgroup $H$ is a key subgroup of a topological group $G$ if there is no strictly coarser Hausdorff group topology on $G$ which induces on $H$ the original topology. In fact, this concept appears implicitly in some earlier publications [9,3,11]. Every co-minimal subgroup is a key subgroup while the converse is not true even for discrete groups. In this paper, we continue the study of minimality in topological matrix groups initiated in [11]. Extending some results from [9,3] concerning the generalized Heisenberg group, we prove that the center of the upper unitriangular group $\operatorname{UT}(n,K)$, defined over a commutative topological unital ring $K$, is a key subgroup. This center is even co-minimal in $\operatorname{UT}(n,K)$ assuming that the multiplication map $m \colon K\times K\to K$ is strongly minimal. We show that the latter holds when $K$ is an archimedean absolute valued field or a local field.