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# Simplicity of Some Twin Tree Automorphism Groups with Trivial Commutation Relations

Published:2014-02-25
Printed: Jun 2014
• Jun Morita,
Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan
• Bertrand Rémy,
Université Lyon 1, Institut Camille Jordan, UMR 5208 du CNRS, 43 blvd du 11 novembre 1918, F-69622 Villeurbanne Cedex, France
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## 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).
 Keywords: Kac-Moody group, twin tree, simplicity, root system, building
 MSC Classifications: 20G44 - Kac-Moody groups 20E42 - Groups with a $BN$-pair; buildings [See also 51E24] 51E24 - Buildings and the geometry of diagrams