Abstract simplicity of complete Kac–Moody groups over finite fields

Let G be a Kac–Moody group over a finite field corresponding to a generalized Cartan matrix A, as constructed by Tits. It is known that G admits the structure of a BN-pair, and acts on its corresponding building. We study the complete Kac–Moody group image which is defined to be the closure of G in the automorphism group of its building. Our main goal is to determine when complete Kac–Moody groups are abstractly simple, that is have no proper non-trivial normal subgroups. Abstract simplicity of image was previously known to hold when A is of affine type. We extend this result to many indefinite cases, including all hyperbolic generalized Cartan matrices A of rank at least four. Our proof uses Tits’ simplicity theorem for groups with a BN-pair and methods from the theory of pro-p groups.