Simply transitive quaternionic lattices of rank 2 over_q(t) and a non-classical fake quadric

By means of a quaternion algebra over $\mathbb{F}$ _q(t), we construct an infinite series of torsion free, simply transitive, irreducible lattices in PGL₂( $\mathbb{F}$ _q((t))) × PGL₂( $\mathbb{F}$ _q((t))). The lattices depend on an odd prime powerq=p^rand a parameter τ ∈ $\mathbb{F}$ _q^×, τ ≠ 1, and are the fundamental group of a square complex with just one vertex and universal coveringT_q+1×T_q+1, a product of trees with constant valencyq+ 1.Our lattices give rise via non-archimedian uniformization to smooth projective surfaces of general type over $\mathbb{F}$ _q((t)) with ample canonical class, Chern numbers c₁²= 2 c₂, trivial Albanese variety and non-reduced Picard scheme. Forq= 3, the Zariski–Euler characteristic attains its minimal value χ = 1: the surface is a non-classical fake quadric.