Cohomological nonvanishing for algebraic fundamental groups of ball quotients

Abstract

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