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.
Residual finiteness for cen- tral extensions of lattices in PU( n, 1) and negatively curved projec- tive varieties
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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.