Automorphisms and opposition in twin buildings

TL;DR

This paper investigates automorphisms of twin buildings, showing they map some residues to opposites and establishing fixed residues for involutions, with implications for finite spherical buildings and projective planes.

Contribution

It proves that automorphisms of twin buildings and finite spherical buildings have fixed residues or map residues to opposites, extending known symmetry properties.

Findings

Automorphisms of twin buildings map some residues to opposites.

No automorphism of certain twin buildings maps all residues of a fixed type to opposites.

Involutions in finite spherical buildings have fixed residues.

Abstract

We show that every automorphism of a thick twin building interchanging the halves of the building maps some residue to an opposite one. Furthermore we show that no automorphism of a locally finite 2-spherical twin building of rank at least 3 maps every residue of one fixed type to an opposite. The main ingredient of the proof is a lemma that states that every duality of a thick finite projective plane admits an absolute point, i.e., a point mapped onto an incident line. Our results also hold for all finite irreducible spherical buildings of rank at least 3, and as a consequence we deduce that every involution of a thick irreducible finite spherical building of rank at least 3 has a fixed residue.