Cohomological nonvanishing for algebraic fundamental groups of ball quotients

Suppose $\mathrm{\Gamma }<\mathrm{P}\mathrm{U}(n,1)$ is a cocompact arithmetic lattice of simplest type with profinite completion $\widehat{\mathrm{\Gamma }}$. This paper proves there is an open subgroup ${\widehat{\mathrm{\Gamma }}}_0\le \widehat{\mathrm{\Gamma }}$ such that $H^j(\widehat{\mathrm{\Delta }},{\mathbb{F}}_p)$ is nontrivial for every open subgroup $\widehat{\mathrm{\Delta }}\le {\widehat{\mathrm{\Gamma }}}_0$, $j\le 2n$, and sufficiently large prime $p$. If $n\ge 2$, nonvanishing is new for all $j\ge 2$. Consequently, the virtual cohomological dimension of $\widehat{\mathrm{\Gamma }}$ 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 $\mathrm{\Gamma }$ and $j<\frac{n+1}{2}$, restriction $H^j(\widehat{\mathrm{\Gamma }},{\mathbb{F}}_p)\to H^j(\mathrm{\Gamma },{\mathbb{F}}_p)$ is shown to be almost surjective in a precise sense; this is related to whether lattices in $\mathrm{P}\mathrm{U}(n,1)$ are good in the sense of Serre, which is only known to hold for $n=1$.