Algebraic & Geometric Topology

On the Chabauty space of locally compact abelian groups

Yves Cornulier

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This paper contains several results about the Chabauty space of a general locally compact abelian group. Notably, we determine its topological dimension, we characterize when it is totally disconnected or connected; we characterize isolated points.

Article information

Algebr. Geom. Topol., Volume 11, Number 4 (2011), 2007-2035.

Received: 6 December 2010
Revised: 8 April 2011
Accepted: 9 April 2011
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22B05: General properties and structure of LCA groups
Secondary: 20E15: Chains and lattices of subgroups, subnormal subgroups [See also 20F22] 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups 54D05: Connected and locally connected spaces (general aspects) 54F45: Dimension theory [See also 55M10]

Chabauty topology locally compact abelian groups


Cornulier, Yves. On the Chabauty space of locally compact abelian groups. Algebr. Geom. Topol. 11 (2011), no. 4, 2007--2035. doi:10.2140/agt.2011.11.2007.

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  • W Banaszczyk, New bounds in some transference theorems in the geometry of numbers, Math. Ann. 296 (1993) 625–635
  • N Bourbaki, Éléments de mathématique. Fascicule XXIX. Livre VI: Intégration. Chapitre 7: Mesure de Haar. Chapitre 8: Convolution et représentations, Actualités Scientifiques et Industrielles 1306, Hermann, Paris (1963)
  • N Bourbaki, Éléments de mathématique. Fascicule XXXII. Théories spectrales. Chapitre I: Algébres normées. Chapitre II: Groupes localement compacts commutatifs, Actualités Scientifiques et Industrielles 1332, Hermann, Paris (1967)
  • N Bourbaki, Éléments de mathématique. Topologie générale. Chapitres 1 à 4, Hermann, Paris (1971)
  • D L Boyer, Enumeration theorems in infinite Abelian groups, Proc. Amer. Math. Soc. 7 (1956) 565–570
  • M R Bridson, P de la Harpe, V Kleptsyn, The Chabauty space of closed subgroups of the three-dimensional Heisenberg group, Pacific J. Math. 240 (2009) 1–48
  • C Chabauty, Limite d'ensembles et géométrie des nombres, Bull. Soc. Math. France 78 (1950) 143–151
  • C Champetier, L'espace des groupes de type fini, Topology 39 (2000) 657–680
  • Y de Cornulier, L Guyot, W Pitsch, On the isolated points in the space of groups, J. Algebra 307 (2007) 254–277
  • Y de Cornulier, L Guyot, W Pitsch, The space of subgroups of an abelian group, J. Lond. Math. Soc. $(2)$ 81 (2010) 727–746
  • J Dixmier, Quelques propriétés des groupes abéliens localement compacts, Bull. Sci. Math. $(2)$ 81 (1957) 38–48
  • R Engelking, Dimension theory, North-Holland Mathematical Library 19, North-Holland Publishing Co., Amsterdam (1978) Translated from the Polish and revised by the author
  • S Fisher, P Gartside, On the space of subgroups of a compact group I, Topology Appl. 156 (2009) 862–871
  • S Fisher, P Gartside, On the space of subgroups of a compact group II, Topology Appl. 156 (2009) 855–861
  • R I Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984) 939–985
  • T Haettel, L'espace des sous-groupes fermés de $\mathbb{R}{\times}\mathbb{Z}$, Algebr. Geom. Topol. 10 (2010) 1395–1415
  • W Hurewicz, H Wallman, Dimension Theory, Princeton Mathematical Series 4, Princeton University Press, Princeton, NJ (1941)
  • J Isbell, Graduation and dimension in locales, from: “Aspects of topology”, London Math. Soc. Lecture Note Ser. 93, Cambridge Univ. Press, Cambridge (1985) 195–210
  • Y Katuta, On the covering dimension of inverse limits, Proc. Amer. Math. Soc. 84 (1982) 588–592
  • B Kloeckner, The space of closed subgroups of $\mathbb{R}^n$ is stratified and simply connected, J. Topol. 2 (2009) 570–588
  • K Mahler, Ein Übertragungsprinzip für konvexe Körper, Časopis P\v est. Mat. Fys. 68 (1939) 93–102
  • L Narens, Topologies of closed subsets, Trans. Amer. Math. Soc. 174 (1972) 55–76
  • B A Pasynkov, The coincidence of various definitions of dimensionality for locally bicompact groups, Dokl. Akad. Nauk SSSR 132 (1960) 1035–1037 Translated as Soviet Math. Dokl. 1 (1960) 720–722
  • I Pourezza, J Hubbard, The space of closed subgroups of $\mathbb{R}^{2}$, Topology 18 (1979) 143–146
  • \=I V Protasov, Dualisms of topological abelian groups, Ukrain. Mat. Zh. 31 (1979) 207–211, 224
  • \=I V Protasov, Y V Tsybenko, Connectedness in the space of subgroups, Ukrain. Mat. Zh. 35 (1983) 382–385
  • I V Protasov, Y V Tsybenko, Chabauty's topology in the lattice of closed subgroups, Ukrain. Mat. Zh. 36 (1984) 207–213
  • P Vopěnka, On the dimension of compact spaces, Czechoslovak Math. J. 8 (83) (1958) 319–327
  • F Wattenberg, Topologies on the set of closed subsets, Pacific J. Math. 68 (1977) 537–551