Cohomological nonvanishing for algebraic fundamental groups of ball quotients

2025/08/28 by Matthew Stover, Stover, Matthew
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2508.20847

Abstract

Suppose Γ< PU(n,1) is a cocompact arithmetic lattice of simplest type with profinite completion \widehatΓ. This paper proves there is an open subgroup \widehatΓ₀ ≤ \widehatΓ such that H^j(\widehatΔ, \mathbbF_p) is nontrivial for every open subgroup \widehatΔ ≤ \widehatΓ₀, j ≤ 2n, and sufficiently large prime p. If n ≥ 2, nonvanishing is new for all j ≥ 2. Consequently, the virtual cohomological dimension of \widehatΓ is at least 2n, improving the previous lower bound of 1. The proof shows there is a profinite fundamental class for the associated ball quotient and that its canonical class is profinite modulo torsion. For congruence Γ and j < (n+1)/(2), restriction H^j(\widehatΓ, \mathbbF_p) → H^j(Γ, \mathbbF_p) is shown to be almost surjective in a precise sense; this is related to whether lattices in PU(n,1) are good in the sense of Serre, which is only known to hold for n=1.

Citations

• Residual finiteness and discrete subgroups of Lie groups
• Products of curves as ball quotients
• Algebraic fundamental groups of fake projective planes
• Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds
• Special subvarieties of non-arithmetic ball quotients and Hodge Theory
• Punctured spheres in complex hyperbolic surfaces and bielliptic ball quotient compactifications
• Bielliptic ball quotient compactifications and lattices in PU(2, 1) with finitely generated commutator subgroup
• The Hodge conjecture and arithmetic quotients of complex balls
• On normal subgroups in the fundamental groups of complex surfaces

Related

• Strongly divisible lattices and crystalline cohomology in the imperfect residue field case
• The Bloch--Kato conjecture, decomposing fields, and generating cohomology in degree one
• Non-algebraic geometrically trivial cohomology classes over finite fields
• Lattices in PU(n,1) that are not profinitely rigid
• Residually finite lattices in \widetildePU(2,1) and fundamental groups of smooth projective surfaces