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.
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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.