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 at least 2 is a
Bruhat--Tits building if and only if its automorphism group acts cocompactly
and chamber-transitively at infinity.Comment: minor typos corrected; reference adde