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Sur les propriétés algébriques et géométriques des groupes de Kac-Moody

Abstract : This text presents a viewpoint from discrete group theory on Kac-Moody groups. Over finite fields, these groups are finitely generated; they act on new buildings with remarkable non-positive curvature properties. We justify that finitely generated Kac-Moody groups can be seen as generalizations of some $S$-arithmetic groups over function fields. We explain how they provide new buildings, and why one can expect that the groups themselves are new. We also consider totally disconnected groups generalizing some semisimple groups over local fields, as illustrated by their combinatorial structure and their topological simplicity. The study of their Furstenberg boundaries is briefly initiated. We sum up the proof of the complete non-linearity of some Kac-Moody groups. This is where we use the properties of the previous topological groups, by combining them with a commensurator super-rigidity theorem. In fact, we can construct groups all of whose linear images are finite, whatever the target field. At last, we make some conjectures about the previously defined groups, for instance the non-linearity (and maybe the abstract simplicity) of a wide class of finitely presented Kac-Moody groups. We also conjecture the abstract simplicity of the geometrically completed Kac-Moody groups, and propose a link between the latter groups and another definition of Kac-Moody groups (constructed as tools in the study of Schubert varieties and in representation theory). We connect these conjectures to work in progress on compactifications of Bruhat-Tits buildings.
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Habilitation à diriger des recherches
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https://tel.archives-ouvertes.fr/tel-00007119
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Submitted on : Thursday, October 14, 2004 - 9:26:30 AM
Last modification on : Wednesday, November 4, 2020 - 2:05:33 PM
Long-term archiving on: : Friday, April 2, 2010 - 8:30:16 PM

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Bertrand Rémy. Sur les propriétés algébriques et géométriques des groupes de Kac-Moody. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2003. ⟨tel-00007119⟩

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