More on $z$-minimality of topological groups

Dekui Peng, Menachem Shlossberg

Published 2023-09-29Version 1

The authors of [4] introduced the notion of $z$-minimality. A minimal group $G$ is $z$-minimal if $G/Z(G)$ is minimal. Answering a question of G. Luk\'acs, we present, in this paper, the $z$-Minimality Criterion for dense subgroups. This criterion implies that if $\Bbb F$ is a subfield of a local field of characteristic distinct than $2,$ then the special linear group $\operatorname{SL}(n,\Bbb F)$ is $z$-minimal precisely when its subgroup $\operatorname{ST^+}(n,\Bbb F)$, the special upper triangular group, is $z$-minimal. Some applications to Number Theory are provided. We prove that the following conditions are equivalent for a positive integer $n$: (a) $n$ is not square-free; (b) there exists a subfield $\mathbb{F}$ of $\mathbb{C}$ such that $\operatorname{SL}(n, \mathbb{F})$ is minimal but not $z$-minimal; (c) there exists a subfield $\mathbb{F}$ of $\mathbb{C}$ such that $\operatorname{ST^+}(n, \mathbb{F})$ is minimal but not $z$-minimal. We also answer [18, Question 6] in the positive, showing that both topological products $G=\prod_{p\in \mathcal P}\operatorname{SL}(p+1,(\mathbb{Q},\tau_p))$ and $H=\prod_{p\in \mathcal P}\operatorname{ST^+}(p+1,(\mathbb{Q},\tau_p))$ are minimal, where $\mathcal P$ is the set of all primes and $(\mathbb{Q},\tau_p)$ is the field of rationals equipped with the $p$-adic topology. In another direction, it is proved that if $G$ is a locally compact $z$-minimal group, then the quotient group $G/Z(G)$ is topologically isomorphic to $\operatorname{Inn}(G),$ the group of all inner automorphisms of $G,$ endowed with the Birkhoff topology.