A fake projective plane is a smooth compact complex surface with the same Betti numbers as $\mathbb{P}^2(\mathbb{C})$. Mumford (MR0527834) built the first one in the late 70s using $p$-adic uniformization. Klingler (MR1990668) and, independently, Yeung (MR2128300 with erratum MR2559112), showed that they necessarily corresponded to arithmetic lattices in $\mathrm{PU}(2,1)$. (That they live in $\mathrm{PU}(2,1)$ follows from Yau's Theorem MR0451180; they do the arithmeticity.) Prasad and Yeung (MR2289867) then gave the list of arithmetic constructions which could possible create such a beast.

Using a computer experiment, Cartwright and Steger gave presentations for all the fundamental groups, giving a complete classification. They have recent announcement in Comptes Rendus #348, 11-13. Their presentations also allow one to show that the congruence subgroup property doesn't hold for certain commensurability classes of lattices for which it was previously unknown. In fact, these lattices are a few of the `simplest' examples for which the congruence subgroup problem remained mysterious. (Serre conjectures in his paper on $\mathrm{SL}_2$, MR0272790, that it fails for all $\mathbb{R}$-rank one groups.)