Some classes of minimally almost periodic topological groups

• Comfort, Wistar ^[1] ; Gould, Franklin R.
  1. [1] Wesleyan University

     Wesleyan University

     Town of Middletown, Estados Unidos

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 16, Nº. 2, 2015, págs. 141-165
• Idioma: inglés
• DOI: 10.4995/agt.2015.3312
• Enlaces
  • Texto completo
• Resumen

  • A Hausdorff topological group G=(G,T) has the small subgroup generating property(briefly: has the SSGP property, or is an SSGP group) if for each neighborhood U of $1_G$ there is a family $\sH$of subgroups of $G$ such that $\bigcup\sH\subseteq U$ and$\langle\bigcup\sH\rangle$ is dense in $G$. The class of $\rm{SSGP}$ groups is defined and investigated with respectto the properties usually studied by topologists (products, quotients, passage to dense subgroups, and the like), and with respect to the familiar class of minimally almostperiodic groups (the m.a.p. groups). Additional classesSSGP(n) for $n˂\omega$ (with SSGP(1) = SSGP) are defined and investigated, and the class-theoretic inclusions$$\mathrm{SSGP}(n)\subseteq\mathrm{SSGP}(n+1)\subseteq\mathrm{ m.a.p.}$$are established and shown proper.In passing the authors also establish the presence of {\rm SSGP}$(1)$ or {\rm SSGP}$(2)$ in many of the early examples in the literature of abelian m.a.p. groups.

• Referencias bibliográficas
  • M. Ajtai, I. Havas and J. Komlós, Every group admits a bad topology, Studies in Pure Mathematics. To the Memory of P. Turán (Paul Erdos, ed.),...
  • W. W. Comfort, Topological groups, In: Handbook of Set-theoretic Topology (Kenneth Kunen and Jerry E. Vaughan, eds.), pp. 1143-1263. North-Holland,...
  • W. W. Comfort and D. Dikranjan, The density nucleus of a topological group, Topology Proc. 44 (2014), 325-356.
  • W. W. Comfort and D. Remus, Long chains of topological group topologies--a continuation, Topology and Its Applications 75 (1997), 51-79. Correction:...
  • Dierolf-Warken, S. Dierolf and S. Warken, Some examples in connection with Pontryagin's duality theorem, Arch. Math. 30 (1978), 599-605.
  • http://dx.doi.org/10.1007/BF01226107
  • D. Dikranjan and D. Shakhmatov,The Markov-Zariski topology of an abelian group, J. Algebra 324 (2010), 1125-1158.
  • http://dx.doi.org/10.1016/j.jalgebra.2010.04.025
  • D. Dikranjan and D. Shakhmatov, Final solution of Protasov-Comfort's problem on minimally almost periodic group topologies, (2014). http://arxiv.org/abs/1410.3313.
  • D. Dikranjan and D. Shakhmatov, Topological groups with many small subgroups, Topology Appl., to appear.
  • R. Ellis and H. B. Keynes, Bohr compactifications and a result of Folner, Israel J. Math. 12 (1972), 314-330.
  • http://dx.doi.org/10.1007/BF02790758
  • L. Fuchs, Infinite Abelian Groups, vol. I, Academic Press, New York-San Francisco-London, 1970.
  • L. Fuchs,Infinite Abelian Groups, vol. II, Academic Press, New York and London, 1973.
  • S. S. Gabriyelyan, Finitely generated subgroups as von Neumann radicals of an Abelian group, Mat. Stud. 38 (2012), 124-138.
  • S. S. Gabriyelyan, Bounded subgroups as a von Neumann radical of an Abelian group, Topology Appl. 178 (2014), 185-199.
  • http://dx.doi.org/10.1016/j.topol.2014.09.007
  • S. S. Gabriyelyan, Minimally almost periodic group topologies on countably infinite abelian groups, Proc. Amer. Math. Soc. 143 (2015), 1823-1829.
  • http://dx.doi.org/10.1090/S0002-9939-2014-12383-5
  • I. M. Gelfand and D. Rauikov, Irreducible unitary representations of locally bicompact groups, Matem. Sbornik N.S. 13 (1943), 301-316. In...
  • E. Glasner, On minimal actions of Polish groups, Topology Appl. 85 (1998), 119-125.
  • http://dx.doi.org/10.1016/S0166-8641(97)00143-0
  • F. R. Gould, An SSGP topology for $ZZ^omega$, Topology Proc. 44 (2014), 389-392
  • M. I. Graev, Free topological groups, In: Topology and Topological Algebra, Translations Series 1, vol. 8, American Mathematical Society,...
  • S. Hartman and J. Mycielski, On the embedding of topological groups into connected topological groups, Colloq. Math. 5 (1958), 167-169.
  • G. Hesse, Zur Topologisierbarkeit von Gruppen, Ph.D. thesis, Universität Hannover, Hannover (Germany), 1979.
  • A. A. Markov, On free topological groups, Doklady Akad. Nauk SSSR 31 (1941), 299-301.
  • A. A. Markov, On free topological groups, In: Topology and Topological Algebra, Translations Series 1, vol. 8, American Mathematical Society,...
  • J. von Neumann, Almost periodic functions in a group I, Trans. Amer. Math. Soc. 36 (1934), 445-492.
  • http://dx.doi.org/10.1090/S0002-9947-1934-1501752-3
  • J. von Neumann and E. P. Wigner, Minimally almost periodic groups, Annals of Math. (Series 2) 41 (1940), 746-750.
  • J. W. Nienhuys, A solenoidal and monothetic minimally almost periodic group, Fund. Math. 73 (1971), 167-169.
  • A. Yu. Ol'shanskiui, A remark on a countable non-topologizable group, Vestnik Mosk. Gos. Univ, Ser. I, Mat. Mekh. 3 (1980), 103. In Russian.
  • V. G. Pestov, Some universal constructions in abstract topological dynamics, In: Topological Dynamics and its Applications. A Volume in Honor...
  • http://dx.doi.org/10.1090/conm/215/02932
  • V. Pestov, Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups., Israel J. Math, 127 (2002), 317-357.
  • http://dx.doi.org/10.1007/BF02784537
  • I. Prodanov, Elementary example of a group without characters, In: Mathematics and Mathematical Education (Sophia, Bulgaria), Bulgarian Acad....
  • I. V. Protasov, Review of [1], Zentralblatt für Matematik 535 (1984), 93.
  • D. Remus,Topological groups without non-trivial characters, In: General Topology and Its Relations to Modern Analysis and Algebra VI (Z. Frolík,...
  • D. Remus, Letter to W. W. Comfort, September, 1989.
  • B. V. Smith-Thomas, Free topological groups, Topology Appl. 4 (1974), 51-72.
  • http://dx.doi.org/10.1016/0016-660X(74)90005-1
  • S. Shelah, On a problem of Kurosh, Jónsson groups and applications, In: Word Problems II (S. I. Adian, W. W. Boone, and G. Higman, eds.),...
  • W. A. Veech, Topological dynamics, Bull. Amer. Math. Soc. 83 (1977), 755-830.
  • http://dx.doi.org/10.1090/S0002-9904-1977-14319-X
  • E. G. Zelenyuk and I. V. Protasov, Topologies on Abelian groups, Math. USSR Izvestiya 37, no. 2 (1991), 445-460.
  • http://dx.doi.org/10.1070/IM1991v037n02ABEH002071