For cocompact arithmetic lattices of simplest type in PU(n,1), the cohomology of the profinite completion is nontrivial up to degree 2n for large primes.
The Hodge con- jecture and arithmetic quotients of complex balls.Acta Math., 216(1):1– 125, 2016
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Cohomological nonvanishing for algebraic fundamental groups of ball quotients
For cocompact arithmetic lattices of simplest type in PU(n,1), the cohomology of the profinite completion is nontrivial up to degree 2n for large primes.