Notes on non-archimedean topological groups

Abstract

We show that the Heisenberg type group H X = (Z 2 ⊕ V ) ⋋ V * , 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) nonarchimedean 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 [8]. We unify some old and new characterization results for non-archimedean groups. We show also that any epimorphism into a non-archimedean group must be dense. Contents 1. Introduction and preliminaries 1 2. Minimality and group representations 3 3. Some facts about non-archimedean groups and uniformities 6 4. The Heisenberg group associated to a Stone space 9 5. More characterizations of non-archimedean groups 11 6. Automorphizable actions and epimorphisms in topological groups 12 References 20