A lattice in a residually non-Desarguesian ${\tilde{A}}_2$-building

We build a building of type ${\tilde{A}}_2$ and a discrete group of automorphisms acting simply transitively on its set of vertices. The characteristic feature of this building is that its rank 2 residues are isomorphic to the Hughes projective plane of order 9, which is non-Desarguesian. This solves a problem asked by W. Kantor in 1986, as well as a question asked by J. Howie in 1989.