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

top
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

top

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

top
  1. Adams J.F., Lectures on Lie Groups, W.A. Benjamin Inc., New York-Amsterdam, 1969, xii+182 pp. Zbl0515.22007MR0252560
  2. Banakh T., Guran I., Protasov I., 10.1016/j.topol.2012.04.010, Topology Appl. 159 (2012), no. 9, 2258–2268. MR2921815DOI10.1016/j.topol.2012.04.010
  3. Bryant R.M., 10.1016/0021-8693(77)90312-X, J. Algebra 48 (1977), no. 2, 340–346. Zbl0408.20022MR0453878DOI10.1016/0021-8693(77)90312-X
  4. Dikranjan D., Giordano Bruno A., 10.4995/agt.2009.1789, Appl. Gen. Topol. 10 (2009), no. 1, 85–119. MR2602604DOI10.4995/agt.2009.1789
  5. Dikranjan D., Shakhmatov D., Selected topics from the structure theory of topological groups, in: Open Problems in Topology II, E. Pearl, ed., Elsevier, 2007, pp. 389–406. 
  6. Dikranjan D., Shakhmatov D., 10.1515/JGT.2008.025, J. Group Theory 11 (2008), no. 3, 421–442. Zbl1167.22001MR2419011DOI10.1515/JGT.2008.025
  7. Dikranjan D., Shakhmatov D., 10.1016/j.jalgebra.2010.04.025, J. Algebra 324 (2010), 1125–1158. Zbl1211.22002MR2671798DOI10.1016/j.jalgebra.2010.04.025
  8. Dikranjan D., Toller D., Markov's problems through the looking glass of Zariski and Markov topologies, Ischia Group Theory 2010, Proc. of the Conference, World Scientific Publ., Singapore, 2011, pp. 87–130. 
  9. Dikranjan D., Toller D., 10.1016/j.topol.2012.05.007, Topology Appl. 159 (2012), no.13, 2951–2972. MR2944768DOI10.1016/j.topol.2012.05.007
  10. Dikranjan D., Toller D., The universal exponents of a group, work in progress. 
  11. Markov A.A., On unconditionally closed sets, Comptes Rendus Dokl. Akad. Nauk SSSR (N.S.) 44 (1944), 180–181 (in Russian). MR0011691
  12. Markov A.A., Izv. Akad. Nauk SSSR, Ser. Mat. 9 (1945), no. 1, 3–64 (in Russian). English translation: A.A. Markov, Three papers on topological groups: I. On the existence of periodic connected topological groups, II. On free topological groups, III. On unconditionally closed sets, Amer. Math. Soc. Transl. 1950, (1950), no. 30, 120 pp. 
  13. Markov A.A., , Mat. Sbornik 18 (1946), 3–28 (in Russian). English translation: A.A. Markov, II. On free topological groups, III. On unconditionally closed sets}, Amer. Math. Soc. Transl. 1950, (1950), no. 30, 120 pp.; another English translation:, Topology and Topological Algebra, Transl. Ser. 1, vol. 8, Amer. Math. Soc., 1962, pp. 273–304. MR0015395
  14. Tkachenko M.G., Yaschenko I., 10.1016/S0166-8641(01)00161-4, Topology Appl. 122 (2002), 435–451. Zbl0997.22003MR1919318DOI10.1016/S0166-8641(01)00161-4
  15. Toller D., Verbal functions of a group, to appear. 

NotesEmbed ?

top

You must be logged in to post comments.