Topological simplicity, commensurator super-rigidity and non-linearities of Kac-Moody groups

2003/02/10 by Bertrand Rémy, Bertrand Remy, Remy, Bertrand +2
Mathematics · #17B67 #22E20 #22E40 #22F50 #51E24 #53C24 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:17B67 #msc:22E20 #msc:22E40 #msc:22F50 #msc:51E24 #msc:53C24

paper · pdf · doi:10.48550/arxiv.math/0302107

30 pages, 2 figures

openalex publication_date 2003/02/10 · arxiv created 2003/02/23 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide new arguments to see topological Kac-Moody groups as generalized semisimple groups over local fields: they are products of topologically simple groups and their Iwahori subgroups are the normalizers of the pro-p Sylow subgroups. We use a dynamical characterization of parabolic subgroups to prove that some countable Kac-Moody groups with Fuchsian buildings are not linear. We show for this that the linearity of a countable Kac-Moody group implies the existence of a closed embedding of the corresponding topological group in a non-Archimedean simple Lie group, thanks to a commensurator super-rigidity theorem proved in the Appendix by P. Bonvin.

Related

• Topological groups of Kac-Moody type, Fuchsian twinnings and their lattices
• Rank one isometries of buildings and quasi-morphisms of Kac-Moody groups
• On some pro-p groups from infinite-dimensional Lie theory
• Kac-Moody groups as discrete groups
• Kac-Moody geometry