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Bounded cohomology is not a profinite invariant
Part of
Daniel Echtler, Holger Kammeyer
Journal:

Canadian Mathematical Bulletin / Volume 67 / Issue 2 / June 2024

Published online by Cambridge University Press:

20 October 2023, pp. 379-390

Print publication:

June 2024
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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.