Commensurators of normal subgroups of lattices

David Fisher; Mahan Mj; Wouter van Limbeek

doi:10.5802/jep.274

Commensurators of normal subgroups of lattices
[Commensurateurs de sous-groupes normaux de réseaux]

David Fisher ¹ ; Mahan Mj ² ; Wouter van Limbeek ³

¹ Department of Mathematics, Rice University, 6100 Main St MS 136, Houston, TX 77005, USA
² School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Navy Nagar, Colaba, Mumbai, Maharashtra 400005, India
³ Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, USA

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1099-1122.

Résumé
Abstract

Nous étudions une question de Greenberg-Shalom concernant l’arithméticité des sous-groupes discrets des groupes de Lie semi-simples avec des commensurateurs denses. Nous répondons positivement à cette question pour les sous-groupes normaux des réseaux. Ceci généralise un résultat du second auteur et de T. Koberda pour certains sous-groupes normaux de réseaux arithmétiques dans $SO (n, 1)$ et $SU (n, 1)$ .

We study a question of Greenberg-Shalom concerning arithmeticity of discrete subgroups of semisimple Lie groups with dense commensurators. We answer this question positively for normal subgroups of lattices. This generalizes a result of the second author and T. Koberda for certain normal subgroups of arithmetic lattices in $SO (n, 1)$ and $SU (n, 1)$ .

Reçu le : 2023-05-27
Accepté le : 2024-08-26
Publié le : 2024-10-08

DOI : 10.5802/jep.274

Classification : 22E40, 20A05, 20G15
Keywords: Discrete subgroups of Lie groups, commensurated subgroups, locally compact groups
Mot clés : Sous-groupes discrets de groupes de Lie, sous-groupes commensurables, groupes localement compacts

Affiliations des auteurs :

David Fisher ¹ ; Mahan Mj ² ; Wouter van Limbeek ³

¹ Department of Mathematics, Rice University, 6100 Main St MS 136, Houston, TX 77005, USA
² School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Navy Nagar, Colaba, Mumbai, Maharashtra 400005, India
³ Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Commensurators of normal subgroups of lattices},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1099--1122},
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David Fisher; Mahan Mj; Wouter van Limbeek. Commensurators of normal subgroups of lattices. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1099-1122. doi : 10.5802/jep.274. https://jep.centre-mersenne.org/articles/10.5802/jep.274/

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