Gromov’s Theory of Multicomplexes with Applications to Bounded Cohomology and Simplicial Volume

Frigerio, Roberto; Moraschini, Marco

doi:10.1090/memo/1402

Cited by 5 publications

(7 citation statements)

References 76 publications

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“…The main ingredients for the proofs are two results from [4]. These results give sufficient conditions for an open cover to imply the finiteness (respectively, the vanishing) of the simplicial volume.…”

Section: Remarkmentioning

confidence: 99%

“…In Section 2, we recall the algebra of inverse sequences, we define the fundamental group at infinity and we specify the terminology regarding open manifolds; in particular, we focus on the classes of tame manifolds, inward tame manifolds, and manifolds which are simply connected at infinity. In Section 3, we define the simplicial volume and we state the results from [4] that will be the key for the proof of Theorem 1, Corollary 2, and Theorem 3, to which Section 4 is devoted.…”

Section: Plan Of the Papermentioning

confidence: 99%

“…We state two results from [4] that give finiteness or vanishing conditions on the simplicial volume in terms of the existence of amenable covers with small multiplicity. First, we give some definitions.…”

Section: Simplicial Volume and Amenable Coversmentioning

confidence: 99%

“…We say that a subset U of a topological space X is amenable if i * (π 1 (U, p)) is an amenable subgroup of π 1 (X, p) for any choice of basepoint p ∈ U, where i : U → X denotes the inclusion. Following [4], we give the following definition. Definition 3.1.…”

Section: Simplicial Volume and Amenable Coversmentioning

confidence: 99%

“…Let M be an open oriented triangulable manifold of dimension m, let W be a neighborhood of infinity, and let U = {U j } j∈N be an open cover of W such that each U j is relatively compact in M. Let us suppose also that U is amenable at infinity and that mult(U ) ≤ m; then ||M|| < +∞. Theorem 3.3 ([4,Corollary 11]). Let M be an open oriented triangulable manifold of dimension m, and let U = {U j } j∈N be an amenable open cover of M (i.e.…”

Section: Simplicial Volume and Amenable Coversmentioning

confidence: 99%

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Simplicial volume of manifolds with amenable fundamental group at infinity

Bargagnati

2024

Glasgow Math. J.

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We show that for $n \neq 1,4$ , the simplicial volume of an inward tame triangulable open $n$ -manifold $M$ with amenable fundamental group at infinity at each end is finite; moreover, we show that if also $\pi _1(M)$ is amenable, then the simplicial volume of $M$ vanishes. We show that the same result holds for finitely-many-ended triangulable manifolds which are simply connected at infinity.

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Section: Remarkmentioning

confidence: 99%

Section: Plan Of the Papermentioning

confidence: 99%

Section: Simplicial Volume and Amenable Coversmentioning

confidence: 99%

Section: Simplicial Volume and Amenable Coversmentioning

confidence: 99%

Section: Simplicial Volume and Amenable Coversmentioning

confidence: 99%

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Simplicial volume of manifolds with amenable fundamental group at infinity

Bargagnati

2024

Glasgow Math. J.

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Normed Amenability and Bounded Cohomology over Non-Archimedean Fields

Fournier-Facio

2024

Memoirs of the AMS

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We study continuous bounded cohomology of totally disconnected locally compact groups with coefficients in a non-Archimedean valued field K \mathbb {K} . To capture the features of classical amenability that induce the vanishing of bounded cohomology with real coefficients, we start by introducing the notion of normed K \mathbb {K} -amenability, of which we prove an algebraic characterization. It implies that normed K \mathbb {K} -amenable groups are locally elliptic, and it relates an invariant, the norm of a K \mathbb {K} -amenable group, to the order of its discrete finite p p -subquotients, where p p is the characteristic of the residue field of K \mathbb {K} . Moreover, we prove a characterization of discrete normed K \mathbb {K} -amenable groups in terms of vanishing of bounded cohomology with coefficients in K \mathbb {K} . The algebraic characterization shows that normed K \mathbb {K} -amenability is a very restrictive condition, so the bounded cohomological one suggests that there should be plenty of groups with rich bounded cohomology with trivial K \mathbb {K} coefficients. We explore this intuition by studying the injectivity and surjectivity of the comparison map, for which surprisingly general statements are available. Among these, we show that if either K \mathbb {K} has positive characteristic or its residue field has characteristic 0 0 , then the comparison map is injective in all degrees. If K \mathbb {K} is a finite extension of Q p \mathbb {Q}_p , we classify unbounded and non-trivial quasimorphisms of a group and relate them to its subgroup structure. For discrete groups, we show that suitable finiteness conditions imply that the comparison map is an isomorphism; this applies in particular to finitely presented groups in degree 2 2 . A motivation as to why the comparison map is often an isomorphism, in stark contrast with the real case, is given by moving to topological spaces. We show that over a non-Archimedean field, bounded cohomology is a cohomology theory in the sense of Eilenberg–Steenrod, except for a weaker version of the additivity axiom which is however equivalent for finite disjoint unions. In particular there exists a Mayer–Vietoris sequence, the main missing piece for computing real bounded cohomology.

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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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Gromov’s Theory of Multicomplexes with Applications to Bounded Cohomology and Simplicial Volume

Cited by 5 publications

References 76 publications

Simplicial volume of manifolds with amenable fundamental group at infinity

Simplicial volume of manifolds with amenable fundamental group at infinity

Normed Amenability and Bounded Cohomology over Non-Archimedean Fields

Central extensions and bounded cohomology

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