Two pairs of simply connected geometries
A chamber system \(C=({\mathcal C},{\mathcal E}_ i,i\in I)\) over I consists of a set \({\mathcal C}\) of objects called chambers, \({\mathcal E}_ i\)- family of partitions of \({\mathcal C}\), where i ranges over I. Let \(C'=({\mathcal C}',{\mathcal E}'\!_ i,i\in I)\) and \(C=({\mathcal C},{\mathcal E}_ i,i\in I)\) be two chamber systems. A morphism \(\phi: C'\to C\) is called a 2-cover if \(\phi\) is surjective and sends every rank 1 or 2 residue of C' bijectively to a rank 1 or 2 residue of C. We say that C' is a universal 2-cover of C if C' is a 2-cover of C with \(\phi: C'\to C\) and if \(\psi: C''\to C\) is another 2-cover, then there is a morphism \(\theta\) : C'\(\to C''\) such that \(\phi =\theta \psi\). If the universal 2-cover of C is isomorphic to C itself, then we say C is 2-connected. The author deals with systems defined by groups. The main result of this paper is a lemma which gives conditions when the chamber system is 2- connected. This is used to prove 2-connectedness in two concrete cases of GABs (geometry that is almost a building).
scientific article; zbMATH DE number 3927790