Irreducible subshifts associated with $\tilde A_2$ buildings

Let $Γ$ be a group of type rotating automorphisms of a building $\cB$ of type $\widetilde A_2$, and suppose that $Γ$ acts freely and transitively on the vertex set of $\cB$. The apartments of $\cB$ 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.

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