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.

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