Open subgroups of locally compact Kac-Moody groups

TL;DR

This paper characterizes open subgroups of complete Kac-Moody groups over finite fields, showing they are closely related to parabolic subgroups and establishing finiteness results using Coxeter group properties.

Contribution

It proves that every open subgroup is contained in a finite index parabolic subgroup and introduces new results on parabolic closures in Coxeter groups.

Findings

Open subgroups are contained in finite index parabolic subgroups.

Finitely many parabolic subgroups contain all open subgroups.

New conditions for parabolic closures in Coxeter groups.

Abstract

Let G be a complete Kac-Moody group over a finite field. It is known that G possesses a BN-pair structure, all of whose parabolic subgroups are open in G. We show that, conversely, every open subgroup of G is contained with finite index in some parabolic subgroup; moreover there are only finitely many such parabolic subgroups. The proof uses some new results on parabolic closures in Coxeter groups. In particular, we give conditions ensuring that the parabolic closure of the product of two elements in a Coxeter group contains the respective parabolic closures of those elements.