We construct pairs of residually finite groups with isomorphic profinite completions such that one has non-vanishing and the other has vanishing real second bounded cohomology. The examples are lattices in different higher rank simple Lie groups. Using Galois cohomology, we actually show that $\operatorname{ SO}^0(n,2)$ for $n \ge 6$ and the exceptional groups $E_{ 6(-14)}$ and $E_{ 7(-25)}$ constitute the complete list of higher rank Lie groups admitting such examples.
Categories: math.GR math.NT