Low-dimensional bounded cohomology and extensions of groups

Heuer, Nicolaus

doi:10.7146/math.scand.a-114969

Cited by 4 publications

(3 citation statements)

References 12 publications

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“…is a central extension and hence corresponds to a class eu ∈ H 2 (Homeo + (S 1 ), Z) the Euler-class. This class is represented by the cocycle ω : Heu17b] for the correspondence of group extensions and bounded cohomology. The image of eu b under the change of coefficients…”

Section: Generalised Quasimorphismsmentioning

confidence: 99%

Gaps in scl for Amalgamated Free Products and RAAGs

Heuer

2019

Geom. Funct. Anal.

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We develop a new criterion to tell if a group G has the maximal gap of 1/2 in stable commutator length (scl). For amalgamated free products G = A C B we show that every element g in the commutator subgroup of G which does not conjugate into A or B satisfies scl(g) ≥ 1/2, provided that C embeds as a left relatively convex subgroup in both A and B. We deduce from this that every non-trivial element g in the commutator subgroup of a right-angled Artin group G satisfies scl(g) ≥ 1/2. This bound is sharp and is inherited by all fundamental groups of special cube complexes.We prove these statements by constructing explicit extremal homogeneous quasimorphisms φ : G → R satisfyingφ(g) ≥ 1 and D(φ) ≤ 1. Such maps were previously unknown, even for non-abelian free groups. For these quasimorphismsφ there is an action ρ : G → Homeo + (S 1 ) on the circle such that [δ 1φ ] = ρ * eu R b ∈ H 2 b (G, R), for eu R b the real bounded Euler class.

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Section: Generalised Quasimorphismsmentioning

confidence: 99%

Gaps in scl for Amalgamated Free Products and RAAGs

Heuer

2019

Geom. Funct. Anal.

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“…Therefore, a group G satisfies Property QITB if and only if the existence of a Lipschitz section for a central extension of G implies the existence of a quasihomomorphic section for the same extension. We refer the reader to [Heu20] for a discussion of (not necessarily central) extensions with bounded Euler class in terms of quasihomomorphisms.…”

Section: The Euler Class Of a Quasi-isometrically Trivial Central Ext...mentioning

confidence: 99%

Central extensions and bounded cohomology

Frigerio¹,

Sisto²

2023

Annales Henri Lebesgue

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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.Finally, we show that Property QITB holds for every finitely presented group if a conjecture by Gromov on bounded primitives of differential forms holds as well.Résumé. -Gersten a montré qu'une extension centrale d'un groupe de type fini est quasiisométriquement triviale lorsque sa classe d'Euler est bornée. On dit qu'un groupe G de type fini vérifie la propriété QITB (quasi-isométriquement trivial implique borné) si la classe d'Euler de toute extension centrale quasi-isométriquement triviale de G est bornée. Nous exhibons un groupe de type fini G qui ne satisfait pas la propriété QITB. Cela répond à une question de Neumann et Reeves et fournit des réponses partielles à des questions reliées de Wienhard et Blank. Nous prouvons également que la propriété QITB est satisfaite par une grande classe de groupes, contenant les groupes moyennables, les groupes d'Artin à angle droit, les groupes

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“…Minimal extensions of approximate groups are closely related to bounded group extensions as defined by Heuer [Heu20]. We briefly recall the relevant definition: DEFINITION 2.45.…”

Section: Extensions Of Approximate Groupsmentioning

confidence: 99%

Foundations of geometric approximate group theory

Cordes¹,

Hartnick²,

Tonić³

2020

Preprint

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We develop the foundations of a geometric theory of countably-infinite approximate groups, extending work of Björklund and the second-named author. Our theory is based on the notion of a quasiisometric quasi-action (qiqac) of an approximate group on a metric space.More specifically, we introduce a geometric notion of finite generation for approximate group and prove that every geometrically finitely-generated approximate group admits a geometric qiqac on a proper geodesic metric space. We then show that all such spaces are quasi-isometric, hence can be used to associate a canonical QI type with every geometrically finitely-generated approximate group. This in turn allows us to define geometric invariants of approximate groups using QI invariants of metric spaces. Among the invariants we consider are asymptotic dimension, finiteness properties, numbers of ends and growth type.A particular focus is on qiqacs on hyperbolic spaces. Our strongest results are obtained for approximate groups which admit a geometric qiqac on a proper geodesic hyperbolic space. For such "hyperbolic approximate groups" we establish a number of fundamental properties in analogy with the case of hyperbolic groups. For example, we show that their asymptotic dimension is one larger than the topological dimension of their Gromov boundary and that -under some mild assumption of being "non-elementary" -they have exponential growth and act minimally on their Gromov boundary. We also show that every non-elementary hyperbolic approximate group of asymptotic dimension 1 is quasi-isometric to a finitelygenerated non-abelian free group.We also study proper qiqacs of approximate group on hyperbolic spaces, which are not necessarily cobounded. Here one focus is on convex cocompact qiqacs, for which we provide several geometric characterizations à la Swenson. In particular we show that all limit points of a convex cocompact qiqac are conical, whereas in general the conical limit set contains additional information.Finally, using the theory of Morse boundaries, we extend some of our results concerning qiqacs on hyperbolic spaces to qiqacs on proper geodesic metric spaces with non-trivial Morse boundary.Throughout this book we emphasize different ways in which definitions and results from geometric group theory can be extended to approximate groups. In cases where there is more than one way to extend a given definition, the relations between different possible generalizations are carefully explored.

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Low-dimensional bounded cohomology and extensions of groups

Cited by 4 publications

References 12 publications

Gaps in scl for Amalgamated Free Products and RAAGs

Gaps in scl for Amalgamated Free Products and RAAGs

Central extensions and bounded cohomology

Foundations of geometric approximate group theory

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