Opposition diagrams for automorphisms of large spherical buildings

J. Parkinson and H. Van Maldeghem

Abstract

Let $\theta$ be an automorphism of a thick irreducible spherical building $\mathrm{\Delta }$ 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 $\mathrm{\Delta }$ mapped onto opposite simplices by $\theta$, then there exists a type $J_1\cup J_2$ simplex of $\mathrm{\Delta }$ mapped onto an opposite simplex by $\theta$. 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.

Keywords: Buildings, projective spaces, polar spaces, domestic automorphism.

AMS Subject Classification: Primary 20E42; secondary 51E24.