Bavard's duality theorem for mixed commutator length

Kawasaki, Morimichi; Kimura, Mitsuaki; Matsushita, Takahiro; Mimura, Mamoru

doi:10.48550/arxiv.2007.02257

Cited by 2 publications

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“…Proposition 2.2 (Proposition 6.4 of [KKMM20]). If the projection p : G → Γ virtually splits, then the map i…”

Section: Preliminariesmentioning

confidence: 99%

“…We introduce the concept of N -quasi-cocycles, which is a generalization of the concept of partial quasimorphisms introduced in [EP06] (see also [MVZ12], [Kaw16], [Kim18], [BK] and [KKMM20]).…”

Section: Cohomology Classes Induced By the Flux Homomorphismmentioning

confidence: 99%

“…To show this theorem, we recall some concepts introduced in [KKMM20]). Using Proposition 6.3, we have the following:…”

Section: Proof Of Theorem 17mentioning

confidence: 99%

“…There are several known conditions that guarantee Q(N ) G = i * Q(G), i.e., every Ginvariant quasimorphism is extendable (see [Sht16], [Ish14], and [KKMM20]). We say that a group homomorphism f : G → Γ virtually splits if there exist a subgroup Λ of finite index of Γ and a group homomorphism s : Λ → G such that f • s(x) = x for every x ∈ Λ.…”

mentioning

confidence: 99%

“…As was mentioned in [KK19] and [KKMM20], it is easy to construct a pair (G, N ) such that scl N and scl G,N are not bi-Lipschitzly equivalent on [N, N ]. Contrastingly, there are only few examples such that scl G and scl G,N are not equivalent on [G, N ].…”

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confidence: 99%

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The space of non-extendable quasimorphisms

Kawasaki¹,

Kimura²,

Maruyama³

et al. 2021

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In the present paper, for a pair (G, N ) of a group G and its normal subgroup N , we consider the space of quasimorphisms and quasi-cocycles on N non-extendable to G. To treat this space, we establish the five-term exact sequence of cohomology relative to the bounded subcomplex. As its application, we study the spaces associated with the commutator subgroup of a Gromov hyperbolic group, the kernel of the (volume) flux homomorphism, and the IA-automorphism group of a free group. Furthermore, we employ this space to prove that the stable commutator length is equivalent to the mixed stable commutator length for certain pairs of a group and its normal subgroup.

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“…Proposition 2.2 (Proposition 6.4 of [KKMM20]). If the projection p : G → Γ virtually splits, then the map i…”

Section: Preliminariesmentioning

confidence: 99%

Section: Cohomology Classes Induced By the Flux Homomorphismmentioning

confidence: 99%

“…To show this theorem, we recall some concepts introduced in [KKMM20]). Using Proposition 6.3, we have the following:…”

Section: Proof Of Theorem 17mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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The space of non-extendable quasimorphisms

Kawasaki¹,

Kimura²,

Maruyama³

et al. 2021

Preprint

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Mixed commutator lengths, wreath products and general ranks

Kawasaki¹,

Kimura²,

Maruyama³

et al. 2022

Preprint

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In the present paper, for a pair (G, N ) of a group G and its normal subgroup N , we consider the mixed commutator length clG,N on the mixed commutator subgroup [G, N ]. We focus on the setting of wreath products:(G, N ) = (Z ≀ Γ, Γ Z). Then we determine mixed commutator lengths in terms of the general rank in the sense of Malcev. In particular, for an abelian group Γ that is not locally cyclic, we show that the ordinary commutator length clG does not coincide with clG,N on [G, N ]. On the other hand, we prove that if Γ is locally cyclic, then for every pair (G, N ) such that 1 → N → G → Γ → 1 is exact, clG and clG,N coincide on [G, N ]. We also study the case of permutational wreath products when the group Γ belongs to a certain class related to surface groups.

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Bavard's duality theorem for mixed commutator length

Cited by 2 publications

References 0 publications

The space of non-extendable quasimorphisms

The space of non-extendable quasimorphisms

Mixed commutator lengths, wreath products and general ranks

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