Bounded Cohomology of Certain Groups of Homeomorphisms

Matsumoto, Shigenori; Morita, Shigeyuki

doi:10.2307/2045250

Cited by 57 publications

(96 citation statements)

References 2 publications

(3 reference statements)

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“…It was proved in [97] that if Γ is a uniformly perfect group, then the mapping H 2 b (Γ )→H 2 (Γ ; R) is injective. Hence if the former part of Conjecture 6.19 were true, then Conjecture 6.21 is also true.…”

Section: Conjecture 621mentioning

confidence: 99%

Structure of the mapping class groups of surfaces: a survey and a prospect

Morita¹

1999

Proceedings of the Kirbyfest

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107

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In this paper, we survey recent works on the structure of the mapping class groups of surfaces mainly from the point of view of topology. We then discuss several possible directions for future research. These include the relation between the structure of the mapping class group and invariants of 3-manifolds, the unstable cohomology of the moduli space of curves and Faber's conjecture, cokernel of the Johnson homomorphisms and the Galois as well as other new obstructions, cohomology of certain infinite dimensional Lie algebra and characteristic classes of outer automorphism groups of free groups and the secondary characteristic classes of surface bundles. We give some experimental results concerning each of them and, partly based on them, we formulate several conjectures and problems.

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Section: Conjecture 621mentioning

confidence: 99%

Structure of the mapping class groups of surfaces: a survey and a prospect

Morita¹

1999

Proceedings of the Kirbyfest

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107

142

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“…A general fact for any group is that in order to deduce the vanishing of H 2 b (−, R) from the vanishing of H 2 (−, R), it suffices to know that the group is uniformly perfect; see e.g. Corollary 2.11 in [18]. Therefore, by Theorem 6.4, we conclude that H 2 b (G, R) vanishes.…”

Section: Acyclicitymentioning

confidence: 92%

Kaleidoscopic groups: permutation groups constructed from dendrite homeomorphisms

2019

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Given a transitive permutation group, a fundamental object for studying its higher transitivity properties is the permutation action of its isotropy subgroup. We reverse this relationship and introduce a universal construction of infinite permutation groups that takes as input a given system of imprimitivity for its isotropy subgroup.This produces vast families of kaleidoscopic groups. We investigate their algebraic properties, such as simplicity and oligomorphy; their homological properties, such as acyclicity or contrariwise large Schur multipliers; their topological properties, such as unique polishability.Our construction is carried out within the framework of homeomorphism groups of topological dendrites.Date: January 2018. B.D. is supported in part by French projects ANR-14-CE25-0004 GAMME and ANR-16-CE40-0022-01 AGIRA. 1 Under the product topology, [n] D n is a Polish space, since D n is countable. The set of colorings C ⊆ [n] D n is easily verified to be a G δ set, so that it is a Polish space under the subspace topology; see for instance [16, Thm. 3.11]. When [n] is finite, C is in fact closed. Definition 3.2. A coloring of D n is kaleidoscopic if for all distinct x, y ∈ Br(D n ) and all distinct i, j ∈ [n], there is z ∈ (x, y) ∩ Br(D n ) such that c z (x) = i and c z (y) = j.In the following picture, open alcoves depict the two components U z (x) and U z (y) respectively colored with i and j.

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“…Since each H λ is amenable, the 1 -seminorm on H n (H λ , R) vanishes (see, for example, [22]), and so, for every λ ∈ Λ, we can choose a representative z λ ∈ Z n (H λ…”

Section: Examples and Counterexamplesmentioning

confidence: 99%

Extending higher-dimensional quasi-cocycles

2015

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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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Bounded Cohomology of Certain Groups of Homeomorphisms

Cited by 57 publications

References 2 publications

Structure of the mapping class groups of surfaces: a survey and a prospect

Structure of the mapping class groups of surfaces: a survey and a prospect

Kaleidoscopic groups: permutation groups constructed from dendrite homeomorphisms

Extending higher-dimensional quasi-cocycles

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