Abstract simplicity of complete Kac–Moody groups over finite fields

Let GG be a Kac–Moody group over a finite field corresponding to a generalized Cartan matrix AA, as constructed by Tits. It is known that GG admits the structure of a BN-pair, and acts on its corresponding building. We study the complete Kac–Moody group Ĝ which is defined to be the closure of GG 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 Ĝ 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 AA of rank at least four. Our proof uses Tits’ simplicity theorem for groups with a BN-pair and methods from the theory of pro-pp groups.