A lattice in a residually non-Desarguesian $\tilde A_2$-building

TL;DR

This paper constructs a specific type of geometric structure called a $ ilde A_2$-building with non-Desarguesian residues, providing a concrete example that addresses longstanding open questions in geometric group theory.

Contribution

It presents the first explicit construction of a $ ilde A_2$-building with Hughes projective plane residues, solving problems posed by Kantor and Howie.

Findings

Constructed a $ ilde A_2$-building with Hughes projective plane residues

Established a discrete automorphism group acting simply transitively on vertices

Provided a counterexample to previous assumptions about such buildings

Abstract

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.