Minimality properties of some topological matrix groups

Abstract

We initiate a systematic study of topological minimality for some matrix groups defined on subfields of local fields. We show that the special upper triangular group $SUT(n,F)$ is minimal for every local field $F$ of characteristic $\neq 2$. This result is new even for the field $\mathbb{R}$ of reals and it leads to some nice consequences. For instance, using Iwasawa decomposition, a new independent proof of the total minimality of the special linear group $SL(n,F)$ is given. This result, which was previously proved by Bader and Gelander (2017), generalizes, in turn, the well-known theorem of Remus and Stoyanov (1991) about the total minimality of $SL(n,\mathbb{R}).$ We provide equivalent conditions for the minimality and total minimality of $SL(n,F)$, where $F$ is a subfield of a local field. In particular, it follows that $SL(2,F)$ is totally minimal and $SL(2^k,F)$ is minimal for every $k$. Extending Remus--Stoyanov theorem in another direction, we show that $SL(n,F)$ is totally minimal for every topological subfield of $\mathbb{R}.$ For some remarkable subfields of local fields we find several, perhaps unexpected, results. Sometimes for the same field, according to the parameter $n,$ we have all three possibilities (a trichotomy): minimality, total minimality and the absence of minimality. We show that if $n$ is not a power of $2$ then $SUT(n,\mathbb{Q}(i))$ and $SL(n,\mathbb{Q}(i))$ are not minimal, where $\mathbb{Q}(i)$ is the Gaussian rational field. Moreover, if $p-1$ is not a power of $2$ then $SL(p-1,(\mathbb{Q},\tau_p))$ is not minimal, where $(\mathbb{Q},\tau_p)$ is the field of rationals equipped with the $p$-adic topology. Furthermore, for every subfield $F$ of $\mathbb{Q}_p,$ the group $SL(n,F)$ is totally minimal for every $n$ which is coprime to $p-1.$Comment: 21 page