| Title:
|
Free non-archimedean topological groups (English) |
| Author:
|
Megrelishvili, Michael |
| Author:
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Shlossberg, Menachem |
| Language:
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English |
| Journal:
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Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
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0010-2628 (print) |
| ISSN:
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1213-7243 (online) |
| Volume:
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54 |
| Issue:
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2 |
| Year:
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2013 |
| Pages:
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273-312 |
| Summary lang:
|
English |
| . |
| Category:
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math |
| . |
| 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]. (English) |
| Keyword:
|
epimorphisms |
| Keyword:
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free profinite group |
| Keyword:
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free topological $G$-group |
| Keyword:
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non-archimedean group |
| Keyword:
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ultra-metric |
| Keyword:
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ultra-norm |
| MSC:
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22A05 |
| MSC:
|
54E15 |
| MSC:
|
54H11 |
| idZBL:
|
Zbl 06221269 |
| idMR:
|
MR3067710 |
| . |
| Date available:
|
2014-07-30T06:08:15Z |
| Last updated:
|
2015-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143275 |
| . |
| Reference:
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