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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
Keywords: Building, symmetric space, CAT(0) space, amenable group, non-positive curvature, locally compact group

Résumé - Abstract

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 ${\ge 2}$ is a Bruhat–Tits building if and only if its automorphism group acts cocompactly and chamber-transitively at infinity.

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