An indiscrete Bieberbach theorem: from amenable CAT$\left(0\right)$ groups to Tits buildings

Caprace, Pierre-Emmanuel; Monod, Nicolas

Journal de l’École polytechnique - Mathématiques, Tome 2 (2015), pp. 333-383.

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.

Non-positively curved spaces admitting a cocompact isometric action of an amenable group are investigated. A classification is established under the assumption that there is no global fixed point at infinity under the full isometry group. The visual boundary is then a spherical building. When the ambient space is geodesically complete, it must be a product of flats, symmetric spaces, biregular trees and Bruhat–Tits buildings.

We provide moreover a sufficient condition for a spherical building arising as the visual boundary of a proper CAT$\left(0\right)$ space to be Moufang, and deduce that an irreducible locally finite Euclidean building of dimension $\ge 2$ is a Bruhat–Tits building if and only if its automorphism group acts cocompactly and chamber-transitively at infinity.

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