Bounded cohomology of finitely presented groups: vanishing, non-vanishing, and computability (Q6609503)

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 0471.20036)]. However, since mitotic groups are far from being finitely generated, the authors apply HNN-extensions to obtain finitely generated groups from mitotic groups, while making sure of preserving boundedly acyclicity in the construction, using a result of \textit{N. Monod} and \textit{S. Popa} [C. R. Math. Acad. Sci., Soc. R. Can. 25, No. 3, 82--87 (2003; Zbl 1040.43001)]. With the same technique, they also prove the existence of a boundedly acyclic non-amenable finitely presented group.\N\NIn the second part of the paper, the authors prove that there exist continuum infinitely many 8-generated groups with large bounded cohomology. The major difficulty they face in proving this is to find a finitely generated group whose bounded cohomology is large in infinitely many degrees. This difficulty is overcome by studying a group introduced by \textit{D. Meier} which is isomorphic to its direct square [J. Lond. Math. Soc., II. Ser. 26, 265--270 (1982; Zbl 0504.20016)]. Moreover, they show that the direct product of the fundamental group of a closed hyperbolic 3-manifold with Thompson's group \(T\) (which is finitely presented) has large bounded cohomology.\N\NFinally, the authors deal with computability of bounded cohomology. It is known that, given a finite group presentation \(\langle S | R \rangle\), where \(S\) is a finite set of generators and \(R\) a finite set of relations, the algorithmic problem of deciding whether \(H_2(\langle S | R \rangle, \mathbb{Z})\) and \(H^2(\langle S | R \rangle, \mathbb{R})\) are trivial or not is undecidable, by a variation of the Adian-Rabin construction [\textit{C. McA. Gordon}, Lond. Math. Soc. Lect. Note Ser. 204, 105--110 (1995; Zbl 0843.20027)]. Similarly, the authors prove that given a finite presentation of a group \(\langle S | R \rangle\) and a natural number \(d \geq 2\), it is undecidable whether\N\N\begin{itemize}\N\item \(H^d_b(\langle S | R \rangle; \mathbb{R})\cong 0\) or not;\N\item \(\dim_{\mathbb{R}}H^d_b(\langle S | R \rangle, \mathbb{R})=|\mathbb{R}|\) or not;\N\item \(\langle S | R \rangle\) is boundedly acylcic or not.\N\end{itemize}\N\NThe same undecidability results are obtained also for bounded cohomology of finite simplicial complexes.