Bounded cohomology is not a profinite invariant

Abstract

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 ${\mathrm{SO}}^0(n,2)$ 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.

Final version to appear in Canad. Math. Bull., 14 pages