Isometric embeddings in bounded cohomology

Bucher, Michelle; Burger, Marc; Frigerio, Roberto; Iozzi, Alessandra; Pagliantini, Cristina; Pozzetti, Maria Beatrice

doi:10.1142/s1793525314500058

Cited by 35 publications

(50 citation statements)

References 52 publications

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“…Then Δ n 1 is adjacent to another simplex Δ n 2 of P . Roughly speaking, we remove from z the singular simplex σ 1 corresponding to Δ n 1 , and modify the singular simplex corresponding to Δ n 2 by suitably 'expanding' it to compensate for removal of σ 1…”

Section: Approximating Real Cycles Via Pseudomanifoldsmentioning

confidence: 99%

“…The only other exact computation of nonvanishing simplicial volume is for the product of two closed hyperbolic surfaces or more generally manifolds locally isometric to the product of two hyperbolic planes [3]. Building on these examples, more values for the simplicial volume can be obtained by surgery or by taking connected sums or amalgamated sums over submanifolds with amenable fundamental groups (see, for example, [1,8,14]); however, not by taking products.…”

Section: Introductionmentioning

confidence: 99%

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The simplicial volume of 3-manifolds with boundary

Bucher

Frigerio

Pagliantini

2015

Journal of Topology

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We provide sharp lower bounds for the simplicial volume of compact 3‐manifolds in terms of the simplicial volume of their boundaries. As an application, we compute the simplicial volume of several classes of 3‐manifolds, including handlebodies and products of surfaces with the interval. Our results provide the first exact computation of the simplicial volume of a compact manifold whose boundary has positive simplicial volume. We also compute the minimal number of tetrahedra in a (loose) triangulation of the product of a surface with the interval.

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Section: Approximating Real Cycles Via Pseudomanifoldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The simplicial volume of 3-manifolds with boundary

Bucher

Frigerio

Pagliantini

2015

Journal of Topology

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“…Our results here are very similar to analogous results proved by Hull and Osin [18], and in fact our arguments were inspired by theirs (even if Hull and Osin's constructions take place in a slightly different context). Perhaps, it is worth mentioning that the exceptional cosets associated to a simplex also generalize the barycenter of the simplex as defined in [6], where the case of amalgamated products is analyzed. Indeed, if G = H * K, then the family {H, K} is hyperbolically embedded in G, and our construction provides a 'quasification' of the strategy described in [6].…”

Section: Quasi-cocycles and Bounded Cohomologymentioning

confidence: 99%

Extending higher-dimensional quasi-cocycles

2015

Self Cite

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Abstract. Let G be a group admitting a non-elementary acylindrical action on a Gromov hyperbolic space (for example, a non-elementary relatively hyperbolic group, or the mapping class group of a closed hyperbolic surface, or Out(Fn) for n ≥ 2). We prove that, in degree 3, the bounded cohomology of G with real coefficients is infinite-dimensional. Our proof is based on an extension to higher degrees of a recent result by Hull and Osin. Namely, we prove that, if H is a hyperbolically embedded subgroup of G and V is any R[G]-module, then any n-quasi cocycle on H with values in V may be extended to G. Also, we show that our extensions detect the geometry of the embedding of hyperbolically embedded subgroups, in a suitable sense.Bounded cohomology of discrete groups is very hard to compute. For example, as observed in [Mon06], there is not a single countable group G whose bounded cohomology (with trivial coefficients) is known in every degree, unless it is known to vanish in all positive degrees (this is the case, for example, of amenable groups). Even worse, no group G is known for which the supremum of the degrees n such that H n b (G, R) = 0 is positive and finite.In degree 2, bounded cohomology has been extensively studied via the analysis of quasi-morphisms (see e.g. [Bro81, EF97, Fuj98, BF02, Fuj00] for the case of trivial coefficients, and [HO13, BBF13] for more general coefficient modules). In this paper we exploit quasi-cocycles, which are the higherdimensional analogue of quasi-morphisms, to prove non-vanishing results for bounded cohomology in degree 3. To this aim, we prove that quasi-cocycles may be extended from a hyperbolically embedded family of subgroups to the ambient group. We also discuss the geometric information carried by our extensions of quasi-cocycles, showing in particular that projections on hyperbolically embedded subgroups may be reconstructed from the extensions of suitably chosen quasi-cocycles.Quasi-cocycles and bounded cohomology. Let G be a group, and V be a normed R[G]-space, i.e. a normed real vector space endowed with an

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“…j factors to a well-defined simplicial mapping 1 . This shows that j is injective on the 1-skeleton.…”

Section: Pairs Of Spacesmentioning

confidence: 99%

“…Bucher et al ( [1,Theorem 1.3]) use other methods from [4] to give an easier proof for the equality M, ∂M = M 1 , ∂M 1 + M 2 , ∂M 2 under the slightly stronger assumption that all components of ∂M have amenable fundamental group. In our result, one only needs the boundary components which are actually glued to be amenable, and also one only needs amenability of im(π 1 A i → π 1 M i ).…”

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confidence: 99%

Multicomplexes, Bounded Cohomology and Additivity of Simplicial Volume

Kuessner¹

2015

Bulletin of the Korean Mathematical Society

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Abstract. We discuss some additivity properties of the simplicial volume for manifolds with boundary: we give proofs of additivity for glueing amenable boundary components and of superadditivity for glueing amenable submanifolds of the boundary, and we discuss doubling of 3-manifolds.This paper is devoted to the behaviour of simplicial volume under cutting and pasting along amenable submanifolds. Such results have been proved by Gromov in [3], for closed manifolds ([3], Section 3) by fairly elementary arguments, and for open manifolds ([3], Section 4) by much more advanced arguments. The aim of this article is to write complete proofs for the case of compact manifolds with boundary, using elementary arguments which are closer to the arguments that were needed for the closed case.The simplicial volume is a homotopy invariant of compact manifolds. It was defined in [3], using the l 1 -norm on singular chains, as follows.

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Isometric embeddings in bounded cohomology

Cited by 35 publications

References 52 publications

The simplicial volume of 3-manifolds with boundary

The simplicial volume of 3-manifolds with boundary

Extending higher-dimensional quasi-cocycles

Multicomplexes, Bounded Cohomology and Additivity of Simplicial Volume

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