A fake plane via 2-adic uniformization with torsion

We adapt the theory of non-Archimedean uniformization to construct a smooth surface from a lattice in PGL3(Q2) that has nontrivial torsion. It turns out to be a fake projective plane, commensurable with Mumford's fake plane yet distinct from it and the other fake planes that arise from 2-adic uniformization by torsion-free groups. As part of the proof, and of independent interest, we compute the homotopy type of the Berkovich space of our plane.