Residual finiteness for central extensions of lattices in \(\mathrm{PU}(n,1)\) and negatively curved projective varieties

• Published in: arXiv.org 2021-12
• Main Authors: Stover, Matthew, Toledo, Domingo
• Format: Article
• Language: English
• Subjects: Lattices; Quotients
• ISSN: 2331-8422

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 \(n \ge 4\).