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

Abstract

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 construct a finitely presented non-amenable boundedly acyclic group. 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~$2$, and a concrete finitely presented one. Countable non-amenable groups with these two extreme properties were previously known to exist, but these constitute the first finitely generated/finitely presented examples. Finally, we show that various algorithmic problems on bounded cohomology are undecidable.