Notes on non-archimedean topological groups

Michael Megrelishvili, Menachem Shlossberg

Published 2010-10-28, updated 2011-06-06Version 2

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.