This paper studies residual finiteness of lattices in the universal cover of and applications to the existence of smooth projective varieties with fundamental group a cocompact lattice in or a finite covering of it. First, we prove that certain lattices in the universal cover of are residually finite. To our knowledge, these are the first such examples. We then use residually finite central extensions of torsion-free lattices in 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 .
To Gopal Prasad in celebration of his 75th birthday
"Residually Finite Lattices in and Fundamental Groups of Smooth Projective Surfaces." Michigan Math. J. 72 559 - 597, August 2022. https://doi.org/10.1307/mmj/20217215