Abstract simplicity of complete Kac-Moody groups over finite fields

Details Abstract simplicity of complete Kac-Moody groups over finite fields Abstract simplicity of complete Kac-Moody groups over finite fields

2006-12-27

arxiv.org/abs/math/0612772v2

Journal of Pure and Applied Algebra 212 (2008), 2147--2162

Mathematics

Group Theory

Final version. The statement and the proof of Theorem 5.2 have been corrected. The main result (Theorem 1.1) now holds under s

Scientific paper

10.1016/j.jpaa.2008.03.023

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 $\hat{G}$ 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 $\hat{G}$ 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.

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