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Pierre-Emmanuel Caprace; Nicolas Monod
An indiscrete Bieberbach theorem: from amenable CAT$(0)$ groups to Tits buildings
(Bieberbach indiscret : des groupes CAT(0) moyennables aux immeubles de Tits)
Journal de l'École polytechnique — Mathématiques, 2 (2015), p. 333-383, doi: 10.5802/jep.26
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Class. Math.: 53C20, 53C24, 43A07, 53C23, 20F65, 20E42
Mots clés: Immeuble, espace symétrique, espace CAT(0), groupe moyennable, courbure négative, groupe localement compact

Résumé - Abstract

Nous étudions les espaces à courbure négative qui admettent une action cocompacte d’un groupe moyennable. Lorsque le groupe de toutes les isométries est sans point fixe global à l’infini, une classification est établie ; le bord à l’infini est alors un immeuble sphérique. Si en outre l’espace est géodésiquement complet, il s’agit nécessairement d’un produit de plats, d’espaces symétriques, d’arbres bi-réguliers et d’immeubles de Bruhat–Tits.

Lorsqu’un immeuble sphérique apparaît comme bord d’un espace CAT(0) propre, nous proposons un critère qui implique la condition de Moufang. Nous en déduisons qu’un immeuble euclidien irréductible localement fini de dimension $\ge 2$ est de Bruhat–Tits si et seulement si son groupe d’automorphismes est cocompact et opère transitivement sur les chambres à l’infini.

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