Local to global structure in twin buildings (Q1910206)

A twin building \(\Delta\) is a pair of buildings \((\Delta_+, \Delta_-)\) with the same Coxeter group \((W, S)\) together with a codistance function \(\delta_* : (\Delta_+ \times \Delta_-) \cup (\Delta_- \times \Delta_+) \to W\) satisfying certain suitable axioms. The axioms imply that there is a natural definition for two chambers \(c_\varepsilon \to \Delta_\varepsilon\) and \(c_{-\varepsilon} \in \Delta_{-\varepsilon}\), \(\varepsilon \in \{+,-\}\), to be opposite. In this way twin buildings can be viewed as generalizations of spherical buildings. One of the main steps in Tits' classification of the thick irreducible spherical buildings of rank at least three is the result that a spherical building is globally determined by its local structure. In the paper under review the authors show that a similar result also holds for twin buildings. They prove that \(\Delta\) is determined by a pair of opposite chambers \((c_+, c_-)\) and the set \(E_2 (c_+)\), provided that \(\Delta_+\) (and \(\Delta_-\)) have no tree residues and that for each \(c \in \Delta_+ \cup \Delta_-\) the set of chambers opposite to \(c\) is connected. This important result might become an essential step towards an eventual classification of all twin buildings.