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.

31 pages