Notes on non-archimedean topological groups

TL;DR

This paper investigates the structure of non-archimedean topological groups, establishing minimality of certain Heisenberg-type groups, and demonstrating how these groups relate to other non-archimedean groups through retractions and actions.

Contribution

It introduces a class of minimal non-archimedean groups derived from Heisenberg-type constructions and unifies various characterization results for non-archimedean groups.

Findings

The Heisenberg-type group $H_X$ is always minimal.

Every non-archimedean group is a retract of a minimal non-archimedean group.

Continuous actions on Stone spaces extend to automorphisms on larger topological groups.

Abstract

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.