Mumford also observed that Yau's result together with Weil's theorem on the rigidity of discrete cocompact subgroups of PU(1,2) implies that there are only a finite number of fake projective planes.
Prasad & Yeung (2007), Prasad & Yeung (2010) found a systematic way of classifying all fake projective planes, by showing that there are twenty-eight classes, each of which contains at least an example of fake projective plane up to isometry, and that there can at most be five more classes which were later shown not to exist.
As a consequence of the work of Aubin and Yau on solution of Calabi Conjecture in the case of negative Ricci curvature, see Yau (1977, 1978), any fake projective plane is the quotient of a complex unit ball in 2 dimensions by a discrete subgroup, which is the fundamental group of the fake projective plane.
Two fake projective planes are defined to be in the same class if their fundamental groups are both contained in the same maximal arithmetic subgroup of automorphisms of the unit ball.
(As a consequence of Prasad & Yeung (2007) and the work of Cartwright and Steger, D has degree 3 over l and the module has dimension 1 over D.) There is one real place of k such that the points of G form a copy of PU(2,1), and over all other real places of k they form the compact group PU(3).
The quotients of the fake projective planes by these groups were studied by Keum (2008) and also by Cartwright & Steger (2010).