Remarks On and Around Bounded Differential Forms

Wienhard, Anna

doi:10.4310/pamq.2012.v8.n2.a5

Cited by 11 publications

(14 citation statements)

References 15 publications

(8 reference statements)

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“…how much freedom one can enjoy in varying the differential representatives of a fixed bounded class. We refer the reader to [BI07,Wie12] for a discussion of this topic. In [KK15] Kim and Kim proved, for example, that if M is a complete, connected, oriented, locally symmetric space of infinite volume, then the Cheeger isoperimetric constant of M is positive if and only if the Riemannian volume form on M admits a bounded primitive.…”

mentioning

confidence: 99%

The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups

Franceschini

Frigerio

Pozzetti

et al. 2019

Comment. Math. Helv.

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We construct combinatorial volume forms of hyperbolic three manifolds fibering over the circle. These forms define non-trivial classes in bounded cohomology. After introducing a new seminorm on exact bounded cohomology, we use these combinatorial classes to show that, in degree 3, the zero norm subspace of the bounded cohomology of an acylindrically hyperbolic group is infinite dimensional. In the appendix we use the same techniques to give a cohomological proof of a lower bound, originally due to Brock, on the volume of the mapping torus of a cobounded pseudo-Anosov homeomorphism of a closed surface in terms of its Teichmüller translation distance.

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confidence: 99%

The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups

Franceschini

Frigerio

Pozzetti

et al. 2019

Comment. Math. Helv.

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“…) is the zero map for every n ∈ N (this was first observed by Gersten). The following question was posed by Wienhard in [Wie12] and by Blank in [Bla15]):…”

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confidence: 99%

“…We refer the reader e.g. [Sik01] for a brief account on the topic, and for a self-contained proof of the fact that bounded classes are d-bounded, and to [BI07,Wie12] for further developments of the theory.…”

Section: ])mentioning

confidence: 99%

“…Question 1 was first asked by Neumann and Reeves in [NR96,NR97] (see also [Why,Remark 2.6]). Moreover, it turns out to be equivalent to questions on ℓ ∞ -cohomology posed in [Wie12,Bla15] (see Question 12 and Proposition 13), and related to a Conjecture by Gromov (see Conjecture 16 and Corollary 18). We provide here a negative answer to Question 1: Theorem 2.…”

Section: Introductionmentioning

confidence: 99%

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Central extensions and bounded cohomology

Frigerio¹,

Sisto²

2020

Preprint

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It was shown by Gersten that a central extension of a finitely generated group is quasi-isometrically trivial provided that its Euler class is bounded. We say that a finitely generated group G satisfies property QITB (quasi-isometrically trivial implies bounded) if the Euler class of any quasi-isometrically trivial central extension of G is bounded. We exhibit a finitely generated group G which does not satisfy Property QITB. This answers a question by Neumann and Reeves, and provides partial answers to related questions by Wienhard and Blank. We also prove that Property QITB holds for a large class of groups, including amenable groups, right-angled Artin groups, relatively hyperbolic groups with amenable peripheral subgroups, and 3-manifold groups.We also show that Property QITB holds for every finitely presented group if and only if a conjecture by Gromov on bounded primitives of differential forms holds as well.

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“…contains the bounded de Rham cohomology class of the n-volume form ω 0 ∈ Ω n (N ); see e.g. [12]. In particular, if N is closed, then this set is exactly the de Rham cohomology class of ω 0 .…”

Section: Introductionmentioning

confidence: 99%

Signed quasiregular curves

Heikkilä¹

2021

Preprint

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We define a subclass of quasiregular curves, called signed quasiregular curves, which contains holomorphic curves and quasiregular mappings. As our main result, we prove a growth theorem of Bonk-Heinonen type for signed quasiregular curves. To obtain our main result, we prove that signed quasiregular curves satisfy a weak reverse Hölder inequality and that this weak reverse Hölder inequality implies the main result. We also obtain higher integrability for signed quasiregular curves. Further, we prove a cohomological value distribution result for signed quasiregular curves by using our main result and equidistribution.

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Remarks On and Around Bounded Differential Forms

Cited by 11 publications

References 15 publications

The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups

The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups

Central extensions and bounded cohomology

Signed quasiregular curves

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