Boundary $C^*$-algebras of triangle geometries

Let $Δ$ be a building of type $\widetilde A_2$ and order $q$, with maximal boundary $Ω$. Let $Γ$ be a group of type preserving automorphisms of $Δ$ which acts regularly on the chambers of $Δ$. Then the crossed product $C^*$-algebra $C(Ω) \rtimes Γ$ is isomorphic to $M_{3(q+1)} \otimes\cl O_{q^2}\otimes\cl O_{q^2}$, where $\cl O_n$ denotes the Cuntz algebra generated by $n$ isometries whose range projections sum to the identity operator.

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