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.
Bounded cohomology of finitely presented groups: Vanishing, non-vanishing and computability / Francesco Fournier-Facio, Clara Löh, Marco Moraschini. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 2036-2145. - ELETTRONICO. - in stampa:(In stampa/Attività in corso), pp. 1-30.
Bounded cohomology of finitely presented groups: Vanishing, non-vanishing and computability
Marco Moraschini
In corso di stampa
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.