  Fundamental groups of fake projective planes fall into fifty distinct
isomorphism classes, one for each complex conjugate pair. We prove that this is
not the case for their algebraic fundamental groups: there are only forty-six
isomorphism classes. We show that there are four pairs of complex conjugate
pairs of fake projective planes that are $\mathrm{Aut}(\mathbb{C})$-equivalent
and hence have mutually isomorphic algebraic fundamental groups. All other
pairs of algebraic fundamental groups are shown to be distinct through explicit
finite \'etale covers. As a by-product, this provides the first examples of
commensurable but nonisomorphic lattices in a rank one semisimple Lie group
that have isomorphic profinite completions.

Algebraic Geometry;Group Theory;Geometric Topology;Number Theory