Abstract
We adapt the theory of non-Archimedean uniformization to construct a smooth surface from a lattice in ${\rm PSL}_3(\mathbb{Q}_2)$ 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.
Citation
Daniel Allcock. Fumiharu Kato. "A fake projective plane via 2-adic uniformization with torsion." Tohoku Math. J. (2) 69 (2) 221 - 237, 2017. https://doi.org/10.2748/tmj/1498269624