THE STRONG RIGIDITY THEOREM FOR NON-ARCHIMEDEAN UNIFORMIZATION

Abstract

In this paper, we present a purely algebraic proof of the strong rigidity for non-Archimedean uniformization, in case the base ring is of characteristic zero. In the last section, we apply this result to Mumford's construction of fake projective planes. In view of recent result on discrete groups by Cartwright, Mantero, Steger and Zappa, we see that there exist at least three fake projective planes.