Algebraic fundamental groups of fake projective planes

TL;DR

This paper investigates the algebraic fundamental groups of fake projective planes, revealing fewer isomorphism classes than their topological counterparts and providing new examples of lattices with isomorphic profinite completions.

Contribution

It proves that the algebraic fundamental groups of fake projective planes form only forty-six classes, fewer than the fifty topological classes, and constructs explicit examples of lattices with isomorphic profinite completions.

Findings

Fewer algebraic fundamental group classes than topological ones.

Identification of four pairs of conjugate fake projective planes with isomorphic algebraic fundamental groups.

First examples of lattices in rank one semisimple Lie groups with isomorphic profinite completions.

Abstract

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.