Bounded cohomology of finitely presented groups: vanishing, non-vanishing, and computability
type: Item
instance of: scholarly article
title: Bounded cohomology of finitely presented groups: vanishing, non-vanishing, and computability
DOI: 10.2422/2036-2145.202201_003
author: Q6047153
author: Clara Löh
author: Marco Moraschini
published in: Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V
publication date: 2024-09-22T00:00:00Z
review text: Bounded cohomology was introduced by Johnson in the 70's in the context of Banach algebras [\textit{B. E. Johnson}, Cohomology in Banach algebras. Providence, RI: American Mathematical Society (AMS) (1972; Zbl 0256.18014)], and was later generalized to topological spaces by \textit{M. Gromov} in the 80's [Publ. Math., Inst. Hautes Étud. Sci. 56, 5--99 (1982; Zbl 0516.53046)]. Despite being widely applied, bounded cohomology is very hard to compute in general. A group whose bounded cohomology vanishes in all positive degree is called \textit{boundedly acyclic}. On the other side of the spectrum, there are groups whose bounded cohomology has dimension greater than or equal to \(|\mathbb{R}|\) in each degree greater than or equal to \(2\): such groups are said to have \textit{large} bounded cohomology. Johnson himself proved that amenable groups are boundedly acyclic. Beyond this class, is very difficult to compute the bounded cohomology of a group in every degree. Even more difficult is to find examples of non-amenable groups whose bounded cohomology is fully understood and which satisfy some finiteness properties, such as being finitely generated or finitely presented. The paper under review gives some results in this direction.\N\NThe main results of the paper concern three different problems:\N\N\begin{itemize}\N\item embedding of finitely generated groups into non-amenable boundedly acyclic finitely generated groups, and existence of finitely presented non-amenable boundedly acyclic groups;\N\item existence of finitely generated groups with large bounded cohomology;\N\item computability of bounded cohomology.\N\end{itemize}\N\NFirst, the authors prove that every finitely generated group embeds in a boundedly acyclic finitely generated non-amenable group. Their proof is based on previous constructions on mitotic groups [\textit{G. Baumslag} et al., J. Pure Appl. Algebra 16, 1--47 (1980; Zbl 0419.20026); Bull. Am. Math. Soc., New Ser. 4, 321--324 (1981; Zbl 0471.200…
reviewed by: Giuseppe Bargagnati
Mathematics Subject Classification ID: 18G90
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 7917524
zbMATH Keywords: bounded cohomology
zbMATH Keywords: boundedly acyclic groups
zbMATH Keywords: mitotic groups
zbMATH Keywords: undecidability
MaRDI profile type: Publication
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