A projection property for buildings

There are two motivations for studying the question whether X has the projection property. Firstly, Arrow's \dictator theorem" states that the only maps of a product X A to X with certain properties are the projections (see A], PRS]). Secondly, the projection property is closely related to the xed point property (see C]). For a detailed discussion of the projection property, its background and connections with other properties see the paper P] in the same volume. In this paper we prove that an irreducible building of spherical type and of rank at least 2 has the projection property. Actually, the theorem is more general. For a precise statement of the hypotheses see below. By contrast, every reducible building and every building of rank one does not have the projection property. We also give a counterexample concerning the niteness assumption of the theorem. I thank M. Pouzet for inspiring discussions and in particular for bringing the notion \projection property" to my attention.