Bounded cohomology of finitely generated groups: vanishing, non-vanishing, and computability

Francesco Fournier-Facio, Clara Loeh, Marco Moraschini

Published 2021-06-25Version 1

We provide new computations in bounded cohomology: A group is boundedly acyclic if its bounded cohomology with trivial real coefficients is zero in all positive degrees. We show that there exists a continuum of finitely generated non-amenable boundedly acyclic groups and that there exists a finitely presented boundedly acyclic group that is universal in the sense that it contains all finitely presented groups. On the other hand, we construct a continuum of finitely generated groups, whose bounded cohomology has uncountable dimension in all degrees greater than or equal to 2. Countable non-amenable groups with these two extreme properties were previously known to exist, but these constitute the first finitely generated examples. Finally, we show that various algorithmic problems on bounded cohomology are undecidable.