The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups

Franceschini, Federico; Frigerio, Roberto; Pozzetti, Maria Beatrice; Sisto, Alessandro

doi:10.4171/cmh/456

Cited by 7 publications

(8 citation statements)

References 35 publications

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“…Here, we are thinking of ∂T α × [0, ∞) as T α away from its core curve. We remark that a very similar construction can be found in §4.4 of [FFPS17], and we are grateful to Maria Beatrice Pozzetti for the observation that it may be helpful for us, here.…”

Section: Infinite Ends Are Cohomologically Separatedsupporting

confidence: 54%

“…In [Som], Soma exhibits a family of linearly independent bounded classes depending on a continuous parameter with arbitrarily small representatives in H 3 b (S; R), thus showing that the zero-norm subspace of bounded cohomology in degree 3 is uncountably generated. This work has recently been generalized in [FFPS17] by Franceschini et al to prove that the zero norm subspace of bounded cohomology in degree 3 of acylindrically hyperbolic groups is uncountably generated.…”

Section: Preliminariesmentioning

confidence: 97%

“…The pseudonorm on degree three bounded cohomology is in general not a norm (see [Som98], [Som], and more recently [FFPS17]). Let ZN 3 (Γ) ⊂ H 3 b (Γ; R) be the subspace consisting of non-trivial classes with zero pseudonorm.…”

Section: Introductionmentioning

confidence: 99%

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Bounded cohomology of finitely generated Kleinian groups

Farre¹

2018

Geom. Funct. Anal.

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Any action of a group Γ on H 3 by isometries yields a class in degree three bounded cohomology by pulling back the volume cocycle to Γ. We prove that the bounded cohomology of finitely generated Kleinian groups without parabolic elements distinguishes the asymptotic geometry of geometrically infinite ends of hyperbolic 3manifolds. That is, if two homotopy equivalent hyperbolic manifolds with infinite volume and without parabolic cusps have different geometrically infinite end invariants, then they define a 2 dimensional subspace of bounded cohomology. Our techniques apply to classes of hyperbolic 3-manifolds that have sufficiently different end invariants, and we give explicit bases for vector subspaces whose dimension is uncountable. We also show that these bases are uniformly separated in pseudo-norm, extending results of Soma. The technical machinery of the Ending Lamination Theorem allows us to analyze the geometrically infinite ends of hyperbolic 3-manifolds with unbounded geometry.

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Section: Infinite Ends Are Cohomologically Separatedsupporting

confidence: 54%

Section: Preliminariesmentioning

confidence: 97%

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Bounded cohomology of finitely generated Kleinian groups

Farre¹

2018

Geom. Funct. Anal.

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“…Proposition 4.15 is essentially equivalent to the fact that the canonical semi-norm in degree 2 is always a norm [42,Corollary 2.7]. This fails already in degree 3 [52,23], but such examples also have large bounded cohomology and so are difficult to control. Therefore we ask: Question 4.17.…”

Section: Proof This Is Essentially a Dual Version Of A Results By Mat...mentioning

confidence: 99%

Bounded cohomology and binate groups

Fournier-Facio,

Loeh,

Moraschini

2021

Preprint

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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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“…[7,25,20,5,15,16,4,32,1,30]), and bounded cohomology in degree 3, which has close ties with the geometry of 3-manifolds (e.g. [7,46,47,48,21,22,17,45,23]). Bounded cohomology in higher degrees, on the contrary, is still largely unexplored.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 92%

Transgression in bounded cohomology and a conjecture of Monod

Ott

2019

J. Topol. Anal.

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We develop an algebro-analytic framework for the systematic study of the continuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology of PSL(2, R) with trivial real coefficients in all degrees greater than two. We prove a vanishing result for strongly reducible classes, thus providing further evidence for a conjecture of Monod. On the cochain level, our method yields explicit formulas for cohomological primitives of arbitrary bounded cocycles. defined by L κ (α) = κ α.Corollary. The bounded Lefschetz map in (1) is zero in all positive degrees.Returning to Theorem 1, we note that in small degrees n = 3, 4, much stronger vanishing theorems apply: Burger and Monod [17] proved that H 3 cb (G; R) = 0, while Hartnick and the author [28] showed that H 4 cb (G; R) = 0. In large degree, on the other hand, our Theorem 1

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The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups

Cited by 7 publications

References 35 publications

Bounded cohomology of finitely generated Kleinian groups

Bounded cohomology of finitely generated Kleinian groups

Bounded cohomology and binate groups

Transgression in bounded cohomology and a conjecture of Monod

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