# Cohomological nonvanishing for algebraic fundamental groups of ball quotients

## TL;DR

This paper proves nonvanishing of cohomology groups for profinite completions of certain arithmetic lattices in complex hyperbolic space, establishing new lower bounds for their virtual cohomological dimension and exploring related profinite classes.

## Contribution

It demonstrates nonvanishing cohomology for all degrees up to 2n in profinite completions of cocompact arithmetic lattices, improving known bounds and introducing profinite fundamental classes.

## Key findings

- Nonvanishing of cohomology for all degrees up to 2n.
- Virtual cohomological dimension at least 2n.
- Almost surjectivity of restriction maps for certain degrees.

## Abstract

Suppose $\Gamma < \mathrm{PU}(n,1)$ is a cocompact arithmetic lattice of simplest type with profinite completion $\widehat{\Gamma}$. This paper proves there is an open subgroup $\widehat{\Gamma}_0 \le \widehat{\Gamma}$ such that $H^j(\widehat{\Delta}, \mathbb{F}_p)$ is nontrivial for every open subgroup ${\widehat{\Delta} \le \widehat{\Gamma}_0}$, $j \le 2n$, and sufficiently large prime $p$. If $n \ge 2$, nonvanishing is new for all $j \ge 2$. Consequently, the virtual cohomological dimension of $\widehat{\Gamma}$ 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 $\Gamma$ and $j < \frac{n+1}{2}$, restriction ${H^j(\widehat{\Gamma}, \mathbb{F}_p) \to H^j(\Gamma, \mathbb{F}_p)}$ 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 $n=1$.