Free non-archimedean topological groups

Michael Megrelishvili; Menachem Shlossberg

Commentationes Mathematicae Universitatis Carolinae (2013)

  • Volume: 54, Issue: 2, page 273-312
  • ISSN: 0010-2628

Abstract

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We study free topological groups defined over uniform spaces in some subclasses of the class 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 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 𝐍𝐀 groups frequently remain continuous. One of the main applications is: any epimorphism in the category 𝐍𝐀 must be dense. Moreover, the same methods improve the following result of T.H. Fay [Fay]: the inclusion of a proper open subgroup H ↪ G ∈ 𝐓𝐆𝐑 is not an epimorphism in the category 𝐓𝐆𝐑 of all Hausdorff topological groups. A key tool in the proofs is Pestov’s test of epimorphisms [V.G. Pestov, Epimorphisms of Hausdorff groups by way of topological dynamics, New Zealand J. Math. 26 (1997), 257–262]. Our results provide a convenient way to produce surjectively universal 𝐍𝐀 abelian and balanced groups. In particular, we unify and strengthen some recent results of Gao [Graev ultrametrics and surjectively universal non-Archimedean Polish groups, Topology Appl. 160 (2013), no. 6, 862–870] and Gao-Xuan [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 [Free pro-C-groups, Math. Z. 125 (1972), 233–254].

How to cite

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Megrelishvili, Michael, and Shlossberg, Menachem. "Free non-archimedean topological groups." Commentationes Mathematicae Universitatis Carolinae 54.2 (2013): 273-312. <http://eudml.org/doc/252529>.

@article{Megrelishvili2013,
abstract = {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 [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, Epimorphisms of Hausdorff groups by way of topological dynamics, New Zealand J. Math. 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 [Graev ultrametrics and surjectively universal non-Archimedean Polish groups, Topology Appl. 160 (2013), no. 6, 862–870] and Gao-Xuan [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 [Free pro-C-groups, Math. Z. 125 (1972), 233–254].},
author = {Megrelishvili, Michael, Shlossberg, Menachem},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {epimorphisms; free profinite group; free topological $G$-group; non-archimedean group; ultra-metric; ultra-norm; -free topological group of a uniform space; non-archimedean group; non-archimedean uniform space; metrizability; epimorphism; automorphizable actions; surjectively universal group},
language = {eng},
number = {2},
pages = {273-312},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Free non-archimedean topological groups},
url = {http://eudml.org/doc/252529},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Megrelishvili, Michael
AU - Shlossberg, Menachem
TI - Free non-archimedean topological groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 2
SP - 273
EP - 312
AB - 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 [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, Epimorphisms of Hausdorff groups by way of topological dynamics, New Zealand J. Math. 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 [Graev ultrametrics and surjectively universal non-Archimedean Polish groups, Topology Appl. 160 (2013), no. 6, 862–870] and Gao-Xuan [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 [Free pro-C-groups, Math. Z. 125 (1972), 233–254].
LA - eng
KW - epimorphisms; free profinite group; free topological $G$-group; non-archimedean group; ultra-metric; ultra-norm; -free topological group of a uniform space; non-archimedean group; non-archimedean uniform space; metrizability; epimorphism; automorphizable actions; surjectively universal group
UR - http://eudml.org/doc/252529
ER -

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