[Paper Review] Topological simplicity, commensurator super-rigidity and non-linearities of Kac-Moody groups

Bertrand Rémy, Patrick Bonvin|arXiv (Cornell University)|Feb 10, 2003

Advanced Operator Algebra Research6 references3 citations

TL;DR

This paper establishes that certain countable Kac-Moody groups over finite fields with right-angled Fuchsian buildings are not linear over any field, using a commensurator super-rigidity theorem and dynamical characterization of parabolic subgroups. It further shows that topological Kac-Moody groups are products of topologically simple groups and that their Iwahori subgroups normalize pro-p Sylow subgroups, reinforcing their analogy to semisimple groups over local fields.

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.

Motivation & Objective

• To prove that certain countable Kac-Moody groups over finite fields are not linear over any field, even in equal characteristic.
• To establish structural parallels between topological Kac-Moody groups and semisimple groups over local fields of positive characteristic.
• To demonstrate that Iwahori subgroups in topological Kac-Moody groups are the normalizers of pro-p Sylow subgroups.
• To provide a dynamical characterization of parabolic subgroups in terms of their action on buildings.
• To extend abstract group homomorphisms to continuous homomorphisms into non-Archimedean Lie groups via super-rigidity.

Proposed method

• Uses a commensurator super-rigidity theorem (proved in an appendix by P. Bonvin) to lift abstract group homomorphisms to continuous homomorphisms into non-Archimedean simple Lie groups.
• Applies dynamical systems techniques, particularly the existence of a $Γ$-equivariant measurable boundary map to homogeneous spaces of the target group.
• Employs the theory of refined Tits systems and virtual pro-p-ness of parahoric subgroups to analyze the structure of topological Kac-Moody groups.
• Utilizes the action of the group on a product of Bruhat-Tits buildings to deduce lattice properties and arithmeticity.
• Applies ergodic theory and Zariski closure arguments to show that boundary maps are not essentially constant, implying non-triviality of the image.
• Uses Zorn’s lemma and descending chain condition on algebraic subgroups to construct a $Λ$-equivariant measurable map from a boundary space to a homogeneous space.

Experimental results

Research questions

• RQ1 Are countable Kac-Moody groups with right-angled Fuchsian buildings linear over any field, including those of equal characteristic?
• RQ2 Can the structure of topological Kac-Moody groups be understood as a generalization of semisimple groups over local fields of positive characteristic?
• RQ3 Is the Iwahori subgroup in a topological Kac-Moody group characterized as the normalizer of a pro-p Sylow subgroup?
• RQ4 Does the existence of a continuous extension of a linear representation imply that the group embeds into a non-Archimedean simple Lie group?
• RQ5 Can dynamical properties of the group action on the building be used to detect non-linearity?

Key findings

• The paper proves that there are infinitely many countable Kac-Moody groups over finite fields of characteristic $p$ that are not linear over any field, for each prime $p$.
• Topological Kac-Moody groups over finite fields are shown to be direct products of topologically simple groups, one for each connected component of the Dynkin diagram.
• Iwahori subgroups in topological Kac-Moody groups are precisely the normalizers of the pro-p Sylow subgroups.
• The linearity of a countable Kac-Moody group implies the existence of a closed embedding into a non-Archimedean simple Lie group, via commensurator super-rigidity.
• The dynamical characterization of parabolic subgroups via boundary maps leads to a contradiction in the linear case, proving non-linearity.
• The boundary map from the Furstenberg boundary to the Grassmannian of a rational representation is not essentially constant, implying that the image is Zariski-dense and non-trivial.