Codistances of 3-spherical buildings

TL;DR

This paper demonstrates that certain 3-spherical buildings with specific connectivity properties admit a twinning structure if they possess a codistance, linking local residue conditions to global twin building structures.

Contribution

It establishes that under particular connectivity conditions, the existence of a codistance guarantees the building is part of a twin building.

Findings

A 3-spherical building with specified residue connectivity admits a twinning.

Presence of a codistance implies the building is half of a twin building.

Connectivity conditions ensure the extension from codistance to twinning.

Abstract

We show that a 3-spherical building in which each rank 2 residue is connected far away from a chamber, and each rank 3 residue is simply 2-connected far away from a chamber, admits a twinning (i.e., is one half of a twin building) as soon as it admits a codistance, i.e., a twinning with a single chamber.