Group Theory/Lie Algebras
New properties of lattices in Lie groups
Comptes Rendus. Mathématique, Volume 338 (2004) no. 4, pp. 271-276.

We study finite extension groups of lattices in Lie groups which have finitely many connected components. We show that every non-cocompact Fuchsian group (these are the non-cocompact lattices in PSL (2,)) has an extension group of finite index which is not isomorphic to a lattice in a Lie group with finitely many connected components. On the other hand we prove that these are, in an appropriate sense, the only lattices in Lie groups which have extension groups of this kind. We also show that an extension group of finite index of a lattice in a Lie group with finitely many connected components has only finitely many conjugacy classes of finite subgroups.

On étudie les extensions finies de réseaux dans les groupes de Lie n'ayant qu'un nombre fini de composantes connexes. Nous démontrons que tout groupe fuchsien (ce sont les réseaux non-cocompacts dans PSL (2,)) possède une extension finie qui n'est isomorphe à aucun réseau dans un groupe de Lie ayant un nombre fini de composantes connexes. D'autre part, nous démontrons que ces groupes sont les seuls, parmi les réseaux dans les groupes de Lie, pour lesquels il existe de telles extensions finies. Nous montrons aussi qu'une extension finie d'un réseau dans un groupe de Lie ayant un nombre fini de composantes connexes n'a qu'un nombre fini de classes de conjugaison de sous-groupes finis.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.12.021
Grunewald, Fritz 1; Platonov, Vladimir 2

1 Mathematisches Institut, Heinrich Heine Universität, 40225 Düsseldorf, Germany
2 Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
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Grunewald, Fritz; Platonov, Vladimir. New properties of lattices in Lie groups. Comptes Rendus. Mathématique, Volume 338 (2004) no. 4, pp. 271-276. doi : 10.1016/j.crma.2003.12.021. http://www.numdam.org/articles/10.1016/j.crma.2003.12.021/

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