On point-line geometries of buildings

Suppose \(\mathcal{B}\) is a building of type \(M=(m_{ij})\), \(\Gamma\) is a Grassmann geometry of \(\mathcal{B}\), and \(G\) is the point-collinearity graph of \(\Gamma\). Many of the properties of the point-line geometry \(\Gamma\) can be deduced from the properties of the chamber system \(\mathcal{B}\).

In particular, \(\Gamma\) is a partial linear space and a gamma space; every singular subspace of \(\Gamma\) is a projective space and is the point shadow of a residue of \(\mathcal{B}\) with diagram is \(\mathsf{A}_k\); every triangle of the point-collinearity graph of \(\Gamma\) is contained in a singular subspace of \(\Gamma\); the shadow of every residues of \(\mathcal{B}\) is a convex subspace of \(\Gamma\). For many spherical building geometries, in particular for all geometries corresponding to a single node of the diagram, the geometry is the convex subspace closure of the shadow of an apartment. Shadows of residues of \(\mathcal{B}\) and of apartments of residues of \(\mathcal{B}\) can be recognized in \(G\).

Let \(\mathcal{C}\) be the set of circular walks in \(\mathcal{B}\) consisting of the circuits of length \(3\) and of the circuits of all possible types \(p_{m_{ij}}(i,j)\), \(\{i,j\} \subseteq I\). Then \(\mathcal{B}\) is \(\mathcal{C}\)-simply connected. Suppose \(X\) is a finite subset of \(\mathcal{C}\) such that

(\(*\))   every arc of \(B\) is traversed the same number of times as its inverse.