Geodesic
Free non-archimedean topological groups
Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 2, pp. 273-312

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

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].
Classification : 22A05, 54E15, 54H11
Keywords: epimorphisms; free profinite group; free topological $G$-group; non-archimedean group; ultra-metric; ultra-norm 
@article{CMUC_2013__54_2_a11,
     author = {Megrelishvili, Michael and Shlossberg, Menachem},
     title = {Free non-archimedean topological groups},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {273--312},
     publisher = {mathdoc},
     volume = {54},
     number = {2},
     year = {2013},
     mrnumber = {3067710},
     zbl = {06221269},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2013__54_2_a11/}
}
TY  - JOUR
AU  - Megrelishvili, Michael
AU  - Shlossberg, Menachem
TI  - Free non-archimedean topological groups
JO  - Commentationes Mathematicae Universitatis Carolinae
PY  - 2013
SP  - 273
EP  - 312
VL  - 54
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CMUC_2013__54_2_a11/
LA  - en
ID  - CMUC_2013__54_2_a11
ER  - 
%0 Journal Article
%A Megrelishvili, Michael
%A Shlossberg, Menachem
%T Free non-archimedean topological groups
%J Commentationes Mathematicae Universitatis Carolinae
%D 2013
%P 273-312
%V 54
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CMUC_2013__54_2_a11/
%G en
%F CMUC_2013__54_2_a11
Megrelishvili, Michael; Shlossberg, Menachem. Free non-archimedean topological groups. Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 2, pp. 273-312. http://geodesic.mathdoc.fr/item/CMUC_2013__54_2_a11/