The space of closed subgroups of a locally compact topological group is endowed with a natural topology, called the Chabauty topology. We completely describe the space of closed subgroups of the group ℝ × ℤ (and of its dual ℂ×), which is highly nontrivial : for example, its fundamental group contains the fundamental group of the Hawaiian earrings, hence is uncountable.
L’espace des sous-groupes fermés d’un groupe topologique localement compact est muni d’une topologie naturelle, appelée topologie de Chabauty. Nous décrivons complètement l’espace des sous-groupes fermés du groupe ℝ × ℤ (et de son dual ℂ×), lequel est hautement non trivial : par exemple, son groupe fondamental contient le groupe fondamental des anneaux Hawaïens, et est donc non dénombrable.
Haettel, Thomas 1
@article{10_2140_agt_2010_10_1395,
author = {Haettel, Thomas},
title = {L{\textquoteright}espace des sous-groupes ferm\'es de {\ensuremath{\mathbb{R}}} {\texttimes} {\ensuremath{\mathbb{Z}}}},
journal = {Algebraic and Geometric Topology},
pages = {1395--1415},
year = {2010},
volume = {10},
number = {3},
doi = {10.2140/agt.2010.10.1395},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2010.10.1395/}
}
Haettel, Thomas. L’espace des sous-groupes fermés de ℝ × ℤ. Algebraic and Geometric Topology, Tome 10 (2010) no. 3, pp. 1395-1415. doi: 10.2140/agt.2010.10.1395
[1] , Éléments de mathématique. Algèbre commutative. Chapitre 10. Profondeur, régularité, dualité, Masson (1998)
[2] , , Metric spaces of non-positive curvature, 319, Springer (1999)
[3] , , , The Chabauty space of closed subgroups of the three-dimensional Heisenberg group, Pacific J. Math. 240 (2009) 1
[4] , , Gromov’s almost flat manifolds, 81, Société Mathématique de France (1981) 148
[5] , , , Notes on notes of Thurston, from: "Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984)" (editor D B A Epstein), London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press (1987) 3
[6] , , The combinatorial structure of the Hawaiian earring group, Topology Appl. 106 (2000) 225
[7] , Limite d’ensembles et géométrie des nombres, Bull. Soc. Math. France 78 (1950) 143
[8] , Pontryagin duality and the Chabauty space of locally compact abelian groups, Article en préparation
[9] , , , Sur la dynamique des groupes de matrices et applications arithmétiques, Éditions de l’École Polytechnique (2007)
[10] , Topology, Allyn and Bacon (1966)
[11] , Spaces of closed subgroups of locally compact groups
[12] , The space of closed subgroups of Rn is stratified and simply connected, J. Topol. 2 (2009) 570
[13] , Topological groups, Gordon and Breach Science Publ. (1966)
[14] , , The space of closed subgroups of R2, Topology 18 (1979) 143
[15] , Cours d’arithmétique, Presses Universitaires de France (1977) 188
[16] , The fundamental group of the Hawaiian earring is not free, Internat. J. Algebra Comput. 2 (1992) 33
Cité par Sources :