Residually Finite Lattices in PU(2,1)˜ and Fundamental Groups of Smooth Projective Surfaces

Abstract

This paper studies residual finiteness of lattices in the universal cover of PU(2,1) and applications to the existence of smooth projective varieties with fundamental group a cocompact lattice in PU(2,1) or a finite covering of it. First, we prove that certain lattices in the universal cover of 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 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 PU(2,1).