Bounded cohomology is not a profinite invariant (Q6634709)

A group invariant is called profinite if it agrees for any two finitely generated residually finite groups \(\Gamma\), \(\Lambda\) with isomorphic profinite completions. A standard example is the abelianization \(H_{1}(0)\). Many group invariants fail to be profinite: property \((\mathsf{T})\), amenability, finiteness properties, Serre's property \(\mathsf{FA}\), Euler characteristic and \(\ell^{2}\)-torsion (see the references in the introduction).\N\NIn the paper under review, the authors 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, they show that \(\mathrm{SO}^{0}(n,2)\) for \(n\geq 6\) and the exceptional groups \(E_{6(-14)}\) and \(E_{7(-25)}\) constitute the complete list of higher-rank Lie groups admitting such examples.\N\NThe reviewer believes it is a very important achievement to produce a complete classification of such counter-examples.