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 $J_1$ and $J_2$ simplices of $Δ$ mapped onto opposite simplices by $θ$, then there exists a type $J_1\cup J_2$ 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.