Irreducible subshifts associated with Ã2 buildings

Let Γ be a group of type rotating automorphisms of a building B of type Ã2, and suppose that Γ acts freely and transitively on the vertex set of B. The apartments of B are tiled by triangles, labelled according to Γ-orbits. Associated with these tilings there is a natural subshift of finite type, which is shown to be irreducible. The key element in the proof is a combinatorial result about finite projective planes.