Abstract simplicity of complete Kac-Moody groups over finite fields

• Autores: Lisa Carbone, Mikhail Ershov, Gordon Ritter
• Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 212, Nº 10, 2008, págs. 2147-2162
• Idioma: inglés
• DOI: 10.1016/j.jpaa.2008.03.023
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• Resumen

  • 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 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 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.