2021
DOI: 10.48550/arxiv.2106.13567
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Bounded cohomology of finitely presented groups: vanishing, non-vanishing, and computability

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 that there exists a finitely presented boundedly acyclic group that is universal in the sense that it contains all finitely presented groups.On the other hand, we construct a continuum of finitely generated groups, whose bounded cohomology has… Show more

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Cited by 5 publications
(16 citation statements)
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“…In particular, every group embeds 2-step subnormally in a boundedly acyclic group. Embeddings into boundedly acyclic groups have been considered before [39,22], but Proposition 4.4 goes one step further, and provides a strong negative answer to Question 4.2: Theorem 4.5. There exists a boundedly acyclic group Γ with a normal subgroup H such that Γ/H is boundedly acyclic, but…”
Section: Hereditary Properties Of Boundedly Acyclic Groupsmentioning
confidence: 99%
See 2 more Smart Citations

Bounded cohomology and binate groups

Fournier-Facio,
Loeh,
Moraschini
2021
Preprint
Self Cite
“…In particular, every group embeds 2-step subnormally in a boundedly acyclic group. Embeddings into boundedly acyclic groups have been considered before [39,22], but Proposition 4.4 goes one step further, and provides a strong negative answer to Question 4.2: Theorem 4.5. There exists a boundedly acyclic group Γ with a normal subgroup H such that Γ/H is boundedly acyclic, but…”
Section: Hereditary Properties Of Boundedly Acyclic Groupsmentioning
confidence: 99%
“…Groups such as H above are said to have large bounded cohomology: Countable [39] and even finitely generated [22] examples are known to exist.…”
Section: Hereditary Properties Of Boundedly Acyclic Groupsmentioning
confidence: 99%
See 1 more Smart Citation

Bounded cohomology and binate groups

Fournier-Facio,
Loeh,
Moraschini
2021
Preprint
Self Cite
“…After providing a positive answer to Question 1.1, we end this introduction by asking a sharper question, where the vanishing of second bounded cohomology is replaced by bounded acyclicity, that is, the vanishing of bounded cohomology with trivial real coefficients in all positive degrees. Several groups of geometric origin have vanishing second bounded cohomology but are not boundedly acyclic [9,Example 3.10].…”
Section: Introductionmentioning
confidence: 99%
“…By Witte-Morris's Theorem, such a group cannot be amenable. The first examples of finitely generated non-amenable boundedly acyclic groups were recently constructed by the first-named author, Löh and Moraschini [9]. However all of these examples are indicable and we do not know whether they can be left orderable.…”
Section: Introductionmentioning
confidence: 99%