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 and discrete subgroups of Lie groups
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Let $G$ be a real Lie group and $\Gamma < G$ be a discrete subgroup of $G$. Is $\Gamma$ residually finite? This paper describes known positive and negative results then poses some questions whose answers will lead to a fairly complete answer for lattices.
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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.