Notes on non-archimedean topological groups

We show that the Heisenberg type group $H_X=(\Bbb{Z}_2 \oplus V) \leftthreetimes V^{\ast}$, with the discrete Boolean group $V:=C(X,\Z_2)$, canonically defined by any Stone space $X$, is always minimal. That is, $H_X$ does not admit any strictly coarser Hausdorff group topology. This leads us to the following result: for every (locally compact) non-archimedean $G$ there exists a (resp., locally compact) non-archimedean minimal group $M$ such that $G$ is a group retract of $M.$ For discrete groups $G$ the latter was proved by S. Dierolf and U. Schwanengel. We unify some old and new characterization results for non-archimedean groups. Among others we show that every continuous group action of $G$ on a Stone space $X$ is a restriction of a continuous group action by automorphisms of $G$ on a topological (even, compact) group $K$. We show also that any epimorphism $f: H \to G$ (in the category of Hausdorff topological groups) into a non-archimedean group $G$ must be dense.Comment: 17 pages, revised versio

Balanced and functionally balanced P-groups

In relation to Itzkowitz\u2019s problem [5], we show that a c-bounded P-group is balanced if and only if it is functionally balanced.We prove that for an arbitrary P-group, being functionally balanced is equivalent to being strongly functionally balanced. A special focus is given to the uniform free topological group defined over a uniform P-space. In particular, we show that this group is (functionally) balanced precisely when its subsets Bn, consisting of words of length at most n, are all (resp., functionally) balanced

Minimality properties of some topological matrix groups

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