Residual finiteness for central extensions of lattices in $\mathrm{PU}(n,1)$ and negatively curved projective varieties
Abstract
We study residual finiteness for cyclic central extensions of cocompact arithmetic lattices $\Gamma < \mathrm{PU}(n,1)$ simple type. We prove that the preimage of $\Gamma$ in any connected cover of $\mathrm{PU}(n,1)$, in particular the universal cover, is residually finite. This follows from a more general theorem on residual finiteness of extensions whose characteristic class is contained in the span in $H^2(\Gamma, \mathbb{Z})$ of the Poincaré duals to totally geodesic divisors on the ball quotient $\Gamma \backslash \mathbb{B}^n$. For $n \ge 4$, if $\Gamma$ is a congruence lattice, we prove residual finiteness of the central extension associated with any element of $H^2(\Gamma, \mathbb{Z})$. Our main application is to existence of cyclic covers of ball quotients branched over totally geodesic divisors. This gives examples of smooth projective varieties admitting a metric of negative sectional curvature that are not homotopy equivalent to a locally symmetric manifold. The existence of such examples is new for all dimensions $n \ge 4$.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.12404
 Bibcode:
 2021arXiv210812404S
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Differential Geometry;
 Mathematics  Group Theory;
 Mathematics  Geometric Topology;
 Mathematics  Number Theory