2021/08/27 by Matthew Stover, Stover, Matthew, Domingo Toledo +1 · 2 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Ball (mathematics) #Combinatorics #Cover (algebra) #Differential Geometry (math.DG) #Dual polyhedron #FOS: Mathematics #Geodesic #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy #Homotopy and Cohomology in Algebraic Topology #Lattice (music) #Mathematical analysis #Mathematics #Number Theory (math.NT) #Physics #Pure mathematics #Quotient
paper · pdf · doi:10.48550/arxiv.2108.12404
openalex publication_date 2021/08/27 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We study residual finiteness for cyclic central extensions of cocompact\narithmetic lattices \Γ < \PU(n,1) simple type. We prove that the\npreimage of \Γ in any connected cover of \PU(n,1), in\nparticular the universal cover, is residually finite. This follows from a more\ngeneral theorem on residual finiteness of extensions whose characteristic class\nis contained in the span in H2(\Γ, \ℤ) of the Poincar 'e duals\nto totally geodesic divisors on the ball quotient \Γ backslash\n mathbbBn. For n \≥ 4, if \Γ is a congruence lattice, we prove\nresidual finiteness of the central extension associated with any element of\nH2(\Γ, \ℤ).\n Our main application is to existence of cyclic covers of ball quotients\nbranched over totally geodesic divisors. This gives examples of smooth\nprojective varieties admitting a metric of negative sectional curvature that\nare not homotopy equivalent to a locally symmetric manifold. The existence of\nsuch examples is new for all dimensions n \≥ 4.\n