Enumerating the fake projective planes

Abstract

A fake projective plane is a smooth compact complex surface \(P\) which is not biholomorphic to the complex projective plane \({\mathbb P}^2_{\mathbb C}\), but has the same Betti numbers as \({\mathbb P}^2_{\mathbb C}\), namely 1, 0, 1, 0, 1. A fake projective plane is determined by its fundamental group.

In their 2007 Inventiones paper, Gopal Prasad and Sai-Kee Yeung showed that these fundamental groups are the torsion-free subgroups \(\Pi\), with finite abelianization, of index \(3/\chi(\bar\Gamma)\) in a maximal arithmetic subgroup \(\bar\Gamma\) of \(\mathrm{PU}(2,1)\). They show that only a small number of \(\bar\Gamma\) can arise, list them explicitly, and found many of the possible subgroups \(\Pi\).

Making heavy use of computers, Tim Steger and I have found all the possible groups \(\Pi\), for all of these \(\bar\Gamma\)s, by finding explicit generators and relations for each of these groups \(\bar\Gamma\). We have therefore found all the fake projective planes. It turns out that there are, up to homeomorphism, exactly 50 of them (100 up to biholomorphism). The fundamental group of Mumford's original fake projective plane will be identified.