Simplicity of some twin tree automorphism groups with trivial commutation relations

TL;DR

This paper proves the simplicity of certain infinite-dimensional Kac-Moody groups over algebraic closures of finite fields, focusing on groups with trivial commutation relations, without relying on the simplicity of their finitely generated counterparts.

Contribution

It establishes the simplicity of incomplete rank 2 Kac-Moody groups over algebraic closures of finite fields with trivial root group relations, independent of the finite field case.

Findings

Proves simplicity of specific Kac-Moody groups

Shows these groups are just infinite modulo center

Provides new methods avoiding finite field simplicity assumptions

Abstract

We prove simplicity for incomplete rank 2 Kac-Moody groups over algebraic closures of finite fields with trivial commutation relations between root groups corresponding to prenilpotent pairs. We don't use the (yet unknown) simplicity of the corresponding finitely generated groups (i.e., when the ground field is finite). Nevertheless we use the fact that the latter groups are just infinite (modulo center).