Minimality of topological matrix groups and Fermat primes

Abstract

Our aim is to study topological minimality of some natural matrix groups. We show that the special upper triangular group $SUT(n,\mathbb{F})$ is minimal for every local field $\mathbb{F}$ of characteristic $\neq 2$. This result is new even for the field $\mathbb{R}$ of reals and it leads to some important consequences. We prove criteria for the minimality and total minimality of the special linear group $SL(n,\mathbb{F})$, where $\mathbb{F}$ is a subfield of a local field. This extends some known results of Remus-Stoyanov (1991) and Bader-Gelander (2017). One of our main applications is a characterization of Fermat primes, which asserts that for an odd prime $p$ the following conditions are equivalent: $\bullet$ $p$ is a Fermat prime; $\bullet$ $SL(p-1,\mathbb{Q})$ is minimal, where $\mathbb{Q}$ is the field of rationals equipped with the $p$-adic topology; $\bullet$ $SL(p-1,\mathbb{Q}(i))$ is minimal, where $\mathbb{Q}(i) \subset \mathbb{C}$ is the Gaussian rational field.