## Tohoku Mathematical Journal 2017 June SECOND SERIES VOL. 69, NO. 2

 Tohoku Math. J. 69 (2017), 221-237

Title A FAKE PROJECTIVE PLANE VIA 2-ADIC UNIFORMIZATION WITH TORSION

Author Daniel Allcock and Fumiharu Kato

(Received December 12, 2014, revised July 27, 2015)
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.

Mathematics Subject Classification. Primary 11F23; Secondary 14J25.

Key words and phrases. Fake projective planes, $p$-adic uniformization.