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

Fournier-Facio, Francesco; Loeh, Clara; Moraschini, Marco

doi:10.48550/arxiv.2106.13567

Cited by 5 publications

(16 citation statements)

References 41 publications

(70 reference 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%

“…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%

“…This class was introduced by Baumslag, Dyer, and Heller [1] to produce embedding results into finitely generated acyclic groups. Bounded acyclicity of mitotic groups, together with co-amenability of ascending HNN extensions, eventually led to finitely generated and finitely presented examples of non-amenable boundedly acyclic groups [22].…”

Section: Introductionmentioning

confidence: 99%

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Bounded cohomology and binate groups

Fournier-Facio,

Loeh,

Moraschini

2021

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A group is boundedly acyclic if its bounded cohomology with trivial real coefficients vanishes in all positive degrees. Amenable groups are boundedly acyclic, while the first non-amenable examples were the group of compactly supported homeomorphisms of R n (Matsumoto-Morita) and mitotic groups (Löh). We prove that binate (alias pseudo-mitotic) groups are boundedly acyclic, which provides a unifying approach to the aforementioned results. Moreover, we show that binate groups are universally boundedly acyclic.We obtain several new examples of boundedly acyclic groups as well as computations of the bounded cohomology of certain groups acting on the circle. In particular, we discuss how these results suggest that the bounded cohomology of the Thompson groups F , T , and V is as simple as possible.

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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%

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Bounded cohomology and binate groups

Fournier-Facio,

Loeh,

Moraschini

2021

Preprint

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“…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%

Finitely generated simple left orderable groups with vanishing second bounded cohomology

Fournier-Facio¹,

Lodha²

2021

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We prove that the finitely generated simple left orderable groups constructed by the second author with Hyde have vanishing second bounded cohomology, both with trivial real and trivial integral coefficients. As a consequence, these are the first examples of finitely generated non-indicable left orderable groups with vanishing second bounded cohomology. This answers Question 8 from the 2018 ICM proceedings article of Andrés Navas. Bounded cohomology and central extensionsWe will work with cohomology and bounded cohomology with trivial real coefficients, and use the definition in terms of the bar resolution. We refer the reader to [3] and [10,22] for a general and complete treatment of ordinary and bounded cohomology, respectively.For every n ≥ 0, denote by C n (G) the set of real-valued functions on G n . By convention, G 0 is a single point, so C 0 (G) ∼ = R consists only of constant functions. We define differential operators

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On the simplicial volume and the Euler characteristic of (aspherical) manifolds

2022

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A well-known question by Gromov asks whether the vanishing of the simplicial volume of oriented closed aspherical manifolds implies the vanishing of the Euler characteristic. We study various versions of Gromov’s question and collect strategies towards affirmative answers and strategies towards negative answers to this problem. Moreover, we put Gromov’s question into context with other open problems in low- and high-dimensional topology. A special emphasis is put on a comparative analysis of the additivity properties of the simplicial volume and the Euler characteristic for manifolds with boundary. We explain that the simplicial volume defines a symmetric monoidal functor (TQFT) on the amenable cobordism category, but not on the whole cobordism category. In addition, using known computations of simplicial volumes, we conclude that the fundamental group of the four-dimensional amenable cobordism category is not finitely generated. We also consider new variations of Gromov’s question. Specifically, we show that counterexamples exist among aspherical spaces that are only homology equivalent to oriented closed connected manifolds.

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

Cited by 5 publications

References 41 publications

Bounded cohomology and binate groups

Bounded cohomology and binate groups

Finitely generated simple left orderable groups with vanishing second bounded cohomology

On the simplicial volume and the Euler characteristic of (aspherical) manifolds

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