Cohomological nonvanishing for algebraic fundamental groups of ball quotients

Suppose Γ< PU(n,1) is a cocompact arithmetic lattice of simplest type with profinite completion Γ. This paper proves there is an open subgroup Γ₀ ≤ Γ such that Hʲ(Δ, Fₚ) is nontrivial for every open subgroup Δ ≤ Γ₀, j ≤ 2n, and sufficiently large prime p. If n ≥ 2, nonvanishing is new for all j ≥ 2. Consequently, the virtual cohomological dimension of Γ 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 Hʲ(Γ, Fₚ) → Hʲ(Γ, Fₚ) 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 $n=1$.