Quaternionic Arithmetic Lattices of Rank 2 and a Fake Quadric in Characteristic 2

Abstract: We construct a torsion-free arithmetic lattice in $\mathrm{PGL}_2(\mathbb{F}_2(!(t)!))\times\mathrm{PGL}_2(\mathbb{F}_2(!(t)!))$ arising from a quaternion algebra over $\mathbb{F}_2(z)$. It is the fundamental group of a square complex with universal covering $T_3\times T_3$, a product of trees with constant valency $3$, which has minimal Euler characteristic. Furthermore, our lattice gives rise to a fake quadric over $\mathbb{F}_2(!(t)!)$ by means of non-archimedean uniformization.