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Keywords:
epimorphisms; free profinite group; free topological $G$-group; non-archimedean group; ultra-metric; ultra-norm
epimorphisms; free profinite group; free topological $G$-group; non-archimedean group; ultra-metric; ultra-norm
Summary:
We study free topological groups defined over uniform spaces in some subclasses of the class $\mathbf{NA}$ of non-archimedean groups. Our descriptions of the corresponding topologies show that for metrizable uniformities the corresponding free balanced, free abelian and free Boolean $\mathbf{NA}$ groups are also metrizable. Graev type ultra-metrics determine the corresponding free topologies. Such results are in a striking contrast with free balanced and free abelian topological groups cases (in standard varieties). Another contrasting advantage is that the induced topological group actions on free abelian $\mathbf{NA}$ groups frequently remain continuous. One of the main applications is: any epimorphism in the category $\mathbf{NA}$ must be dense. Moreover, the same methods improve the following result of T.H. Fay \cite{Fay}: the inclusion of a proper open subgroup $H\hookrightarrow G\in \mathbf{TGR}$ is not an epimorphism in the category $\mathbf{TGR}$ of all Hausdorff topological groups. A key tool in the proofs is Pestov's test of epimorphisms [V.G. Pestov, {\it Epimorphisms of Hausdorff groups by way of topological dynamics\/}, New Zealand J. Math. {\bf 26} (1997), 257--262]. Our results provide a convenient way to produce surjectively universal $\mathbf{NA}$ abelian and balanced groups. In particular, we unify and strengthen some recent results of Gao [{\it Graev ultrametrics and surjectively universal non-Archimedean Polish groups\/}, Topology Appl. {\bf 160} (2013), no. 6, 862--870] and Gao-Xuan [{\it On non-Archimedean Polish groups with two-sided invariant metrics\/}, preprint, 2012] as well as classical results about profinite groups which go back to Iwasawa and Gildenhuys-Lim [{\it Free pro-C-groups\/}, Math. Z. {\bf 125} (1972), 233--254].
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