VSE knjižnice (vzajemna bibliografsko-kataložna baza podatkov COBIB.SI)
  • The Markov and Zariski topologies of some linear groups
    Dikranjan, Dikran N., 1950- ; Toller, Daniele
    We study the Zariski topology ▫$\mathfrak{Z}_G$▫, the Markov topology ▫$\mathfrak{M}_G$▫ and the precompact Markov topology ▫$\mathfrak{P}_G$▫ of an infinite group ▫$G$▫, introduced in Dikranjan and ... Shakhmatov: [D. Dikranjan, D. Shakhmatov, Selected topics from the structure theory of topological groups, in: E. Perl (Ed.), Open Problems in Topology II, Elsevier, 2007, pp. 389-406],[D. Dikranjan, D. Shakhmatov, Reflection principle characterizing groups in which unconditionally closed sets are algebraic, J. Group Theory 11 (3) (2008) 421-442] and [D. Dikranjan, D. Shakhmatov, The Markov-Zariski topology of an abelian group, J. Algebra 324 (2010) 1125-1158]. We prove that ▫$\mathfrak{P}_G$▫ is discrete for a non-abelian divisible solvable group ▫$G$▫, concluding that a countable divisible solvable group ▫$G$▫ is abelian if and only if ▫$\mathfrak{M}_G = \mathfrak{P}_G$▫ if and only if ▫$\mathfrak{P}_G$▫ is non-discrete. This answers Dikranjan and Shakhmatov [D. Dikranjan, D. Shakhmatov, The Markov-Zariski topology of an abelian group, J. Algebra 324 (2010) 1125-1158, Question 12.1]. We study in detail the space ▫$(G, \mathfrak{Z}_G)$▫ for two types of linear groups, obtaining a complete description of various topological properties (as dimension, Noetherianity, etc.). This allows us to distinguish, in the case of linear groups, the Zariski topology defined via words (i.e., the verbal topology in terms of Bryant) from the affine topology usually considered in algebraic geometry. We compare the properties of the Zariski topology of these linear groups with the corresponding ones obtained in Dikranjan and Shakhmatov [D. Dikranjan, D. Shakhmatov, The Markov-Zariski topology of an abelian group, J. Algebra 324 (2010) 1125-1158] in the case of abelian groups.
    Vir: Proceedings of the Eleventh Prague Symposium on General Topology and its Relations to Modern Analysis and Algebra, TOPOSYM 2011 : special issue (Str. 2951-2972)
    Vrsta gradiva - prispevek na konferenci
    Leto - 2012
    Jezik - angleški
    COBISS.SI-ID - 16338777

    Povezava(-e):

    http://dx.doi.org/10.1016/j.topol.2012.05.007

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