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

Pierre-Emmanuel Caprace; Nicolas Monod

doi:10.5802/jep.26

An indiscrete Bieberbach theorem: from amenable CAT

(0)

groups to Tits buildings
[Bieberbach indiscret : des groupes CAT(0) moyennables aux immeubles de Tits]

Pierre-Emmanuel Caprace ¹ ; Nicolas Monod ²

¹ Université catholique de Louvain, IRMP Chemin du Cyclotron 2, bte L7.01.02, 1348 Louvain-la-Neuve, Belgique
² EPFL 1015 Lausanne, Switzerland

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

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 $\geq 2$ 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 $(0)$ space to be Moufang, and deduce that an irreducible locally finite Euclidean building of dimension $\geq 2$ is a Bruhat–Tits building if and only if its automorphism group acts cocompactly and chamber-transitively at infinity.

Reçu le : 2015-03-25
Accepté le : 2015-09-22

MR Zbl

DOI : 10.5802/jep.26

Classification : 53C20, 53C24, 43A07, 53C23, 20F65, 20E42
Keywords: Building, symmetric space, CAT(0) space, amenable group, non-positive curvature, locally compact group
Mot clés : Immeuble, espace symétrique, espace CAT(0), groupe moyennable, courbure négative, groupe localement compact

Affiliations des auteurs :

Pierre-Emmanuel Caprace ¹ ; Nicolas Monod ²

¹ Université catholique de Louvain, IRMP Chemin du Cyclotron 2, bte L7.01.02, 1348 Louvain-la-Neuve, Belgique
² EPFL 1015 Lausanne, Switzerland

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Pierre-Emmanuel Caprace and Nicolas Monod},
     title = {An indiscrete {Bieberbach} theorem: from~amenable {CAT}$(0)$ groups to {Tits} buildings},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {333--383},
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Pierre-Emmanuel Caprace; Nicolas Monod. An indiscrete Bieberbach theorem: from amenable CAT$(0)$ groups to Tits buildings. Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 333-383. doi : 10.5802/jep.26. https://jep.centre-mersenne.org/articles/10.5802/jep.26/

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