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 \(\Delta\) 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 \(\Delta\) mapped onto opposite simplices by \(\theta\), then there exists a type \(J_1\cup J_2\) simplex of \(\Delta\) 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.