Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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F. Fournier-Facio - C. Loeh - M. Moraschini

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

created by moraschini on 05 Jul 2021
modified on 09 Jan 2023


Accepted Paper

Inserted: 5 jul 2021
Last Updated: 9 jan 2023

Journal: Ann. Sc. Norm. Super. Pisa Cl. Sci.
Year: 2022

ArXiv: 2106.13567 PDF


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 generatedfinitely presented examples.

Finally, we show that various algorithmic problems on bounded cohomology are undecidable.

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