Opposition diagrams for automorphisms of large spherical buildings

Abstract

Let θ be an automorphism of a thick irreducible spherical building Δ of rank at least 3 with no Fano plane residues. We prove that if there exist both type J1 and J2 simplices of Δ mapped onto opposite simplices by θ, then there exists a type J1∪J2 simplex of Δ mapped onto an opposite simplex by θ. This property is called cappedness. We give applications of cappedness to opposition diagrams, domesticity, and the calculation of displacement in spherical buildings. In a companion piece to this paper we study the thick irreducible spherical buildings containing Fano plane residues. In these buildings automorphisms are not necessarily capped.