Opposition diagrams for automorphisms of large spherical buildings

TL;DR

This paper proves a property called cappedness for automorphisms of large spherical buildings, showing how oppositeness of certain simplices implies the existence of larger oppositeness structures, with applications to opposition diagrams and displacement calculations.

Contribution

It establishes the cappedness property for automorphisms of large spherical buildings without Fano plane residues, advancing understanding of their symmetry structures.

Findings

Cappedness holds for automorphisms with opposite simplices of certain types.

Applications to opposition diagrams and displacement calculations.

Automorphisms in buildings with Fano residues are not necessarily capped.

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.