Irreducible subshifts associated with ${\tilde{A}}_2$ buildings

Let $\mathrm{\Gamma }$ be a group of type rotating automorphisms of a building $\text{cB}$ of type ${\tilde{A}}_2$, and suppose that $\mathrm{\Gamma }$ acts freely and transitively on the vertex set of $\text{cB}$. The apartments of $\text{cB}$ are tiled by triangles, labelled according to $\mathrm{\Gamma }$-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.