Residually finite lattices in $\widetilde{\mathrm{PU}(2,1)}$ and fundamental groups of smooth projective surfaces

TL;DR

This paper proves residual finiteness for certain lattices in the universal cover of PU(2,1) and constructs smooth projective surfaces with specific fundamental groups, advancing understanding of their geometric and group-theoretic properties.

Contribution

It provides the first examples of residually finite lattices in the universal cover of PU(2,1) and constructs new smooth projective surfaces with prescribed fundamental groups.

Findings

Certain lattices in the universal cover of PU(2,1) are residually finite.

Constructed smooth projective surfaces with fundamental groups as cocompact lattices in PU(2,1).

Showed these surfaces are not birationally equivalent to ball quotients.

Abstract

This paper studies residual finiteness of lattices in the universal cover of $\mathrm{PU}(2,1)$ and applications to the existence of smooth projective varieties with fundamental group a cocompact lattice in $\mathrm{PU}(2,1)$ or a finite covering of it. First, we prove that certain lattices in the universal cover of $\mathrm{PU}(2,1)$ are residually finite. To our knowledge, these are the first such examples. We then use residually finite central extensions of torsion-free lattices in $\mathrm{PU}(2,1)$ to construct smooth projective surfaces that are not birationally equivalent to a smooth compact ball quotient but whose fundamental group is a torsion-free cocompact lattice in $\mathrm{PU}(2,1)$.