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

Abstract

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