Productivity of the Zariski topology on groups

Dikran N. Dikranjan; D. Toller

Commentationes Mathematicae Universitatis Carolinae (2013)

  • Volume: 54, Issue: 2, page 219-237
  • ISSN: 0010-2628

Abstract

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This paper investigates the productivity of the Zariski topology ℨ G of a group G . If 𝒢 = { G i ∣ i ∈ I } is a family of groups, and G = ∏ i ∈ I G i is their direct product, we prove that ℨ G ⊆ ∏ i ∈ I ℨ G i . This inclusion can be proper in general, and we describe the doubletons 𝒢 = { G 1 , G 2 } of abelian groups, for which the converse inclusion holds as well, i.e., ℨ G = ℨ G 1 × ℨ G 2 . If e 2 ∈ G 2 is the identity element of a group G 2 , we also describe the class Δ of groups G 2 such that G 1 × { e 2 } is an elementary algebraic subset of G 1 × G 2 for every group G 1 . We show among others, that Δ is stable under taking finite products and arbitrary powers and we describe the direct products that belong to Δ . In particular, Δ contains arbitrary direct products of free non-abelian groups.

How to cite

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Dikranjan, Dikran N., and Toller, D.. "Productivity of the Zariski topology on groups." Commentationes Mathematicae Universitatis Carolinae 54.2 (2013): 219-237. <http://eudml.org/doc/252537>.

@article{Dikranjan2013,
abstract = {This paper investigates the productivity of the Zariski topology $\mathfrak \{Z\}_G$ of a group $G$. If $\mathcal \{G\} = \lbrace G_i\mid i\in I\rbrace $ is a family of groups, and $G = \prod _\{i\in I\}G_i$ is their direct product, we prove that $\mathfrak \{Z\}_G\subseteq \prod _\{i\in I\}\mathfrak \{Z\}_\{G_i\}$. This inclusion can be proper in general, and we describe the doubletons $\mathcal \{G\} = \lbrace G_1,G_2\rbrace $ of abelian groups, for which the converse inclusion holds as well, i.e., $\mathfrak \{Z\}_G = \mathfrak \{Z\}_\{G_1\}\times \mathfrak \{Z\}_\{G_2\}$. If $e_2\in G_2$ is the identity element of a group $G_2$, we also describe the class $\Delta $ of groups $G_2$ such that $G_1\times \lbrace e_\{2\}\rbrace $ is an elementary algebraic subset of $\{G_1\times G_2\}$ for every group $G_1$. We show among others, that $\Delta $ is stable under taking finite products and arbitrary powers and we describe the direct products that belong to $\Delta $. In particular, $\Delta $ contains arbitrary direct products of free non-abelian groups.},
author = {Dikranjan, Dikran N., Toller, D.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Zariski topology; (elementary; additively) algebraic subset; $\delta $-word; universal word; verbal function; (semi) $\mathfrak \{Z\}$-productive pair of groups; direct product; productivity of Zariski topology; direct products; elementary algebraic subsets; additively algebraic subsets; -words; universal words; verbal functions; productive pairs of groups},
language = {eng},
number = {2},
pages = {219-237},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Productivity of the Zariski topology on groups},
url = {http://eudml.org/doc/252537},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Dikranjan, Dikran N.
AU - Toller, D.
TI - Productivity of the Zariski topology on groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 2
SP - 219
EP - 237
AB - This paper investigates the productivity of the Zariski topology $\mathfrak {Z}_G$ of a group $G$. If $\mathcal {G} = \lbrace G_i\mid i\in I\rbrace $ is a family of groups, and $G = \prod _{i\in I}G_i$ is their direct product, we prove that $\mathfrak {Z}_G\subseteq \prod _{i\in I}\mathfrak {Z}_{G_i}$. This inclusion can be proper in general, and we describe the doubletons $\mathcal {G} = \lbrace G_1,G_2\rbrace $ of abelian groups, for which the converse inclusion holds as well, i.e., $\mathfrak {Z}_G = \mathfrak {Z}_{G_1}\times \mathfrak {Z}_{G_2}$. If $e_2\in G_2$ is the identity element of a group $G_2$, we also describe the class $\Delta $ of groups $G_2$ such that $G_1\times \lbrace e_{2}\rbrace $ is an elementary algebraic subset of ${G_1\times G_2}$ for every group $G_1$. We show among others, that $\Delta $ is stable under taking finite products and arbitrary powers and we describe the direct products that belong to $\Delta $. In particular, $\Delta $ contains arbitrary direct products of free non-abelian groups.
LA - eng
KW - Zariski topology; (elementary; additively) algebraic subset; $\delta $-word; universal word; verbal function; (semi) $\mathfrak {Z}$-productive pair of groups; direct product; productivity of Zariski topology; direct products; elementary algebraic subsets; additively algebraic subsets; -words; universal words; verbal functions; productive pairs of groups
UR - http://eudml.org/doc/252537
ER -

References

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  10. Dikranjan D., Toller D., The universal exponents of a group, work in progress. 
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  15. Toller D., Verbal functions of a group, to appear. 

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