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

TL;DR

This paper advances the understanding of bounded cohomology in finitely presented groups by constructing examples with extreme properties and proving the undecidability of related algorithmic problems.

Contribution

It provides the first finitely generated and finitely presented examples of groups with extreme bounded cohomology properties and establishes the undecidability of certain computational problems.

Findings

Existence of a continuum of non-amenable boundedly acyclic groups.

Construction of a finitely presented non-amenable boundedly acyclic group.

Undecidability of algorithmic problems in bounded cohomology.

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.